Of specific tube BE to the slit at B. The ground-glass cover is then placed air-tight upon the mouth A. There is now no air in the tube AE, except that which is mixed with the sand in the tube AB. If we now suppose that the atmospheric pressure indicated by the barometer is 30 inches, and elevate the tube AE till the mercury stands within it at a point C, 15 inches (one-half of 30), above its surface in the open vessel F, it is manifest that the air within the tube is pressed with exactly half an atmosphere, and therefore expands to twice its original bulk. Owing to this expansion of the air to twice its bulk, the tube AB contains only half the quantity of air which it did at first; and as the part BC contains the other half, the quantity of air in AB and BC is equal, as the air in BC is equal to the air that is mixed with the sand in AB, and fills the same space which the whole occupied previous to its expansion.

Let the sand be now removed from AB, and the same experiment repeated when the tube AB is filled with air. The quantity of air being now greater, will, when expanded to twice its bulk under a pressure of 15 inches, fill a larger space than BC, and the mercury will rise only to some point D. But as the expanded air in E occupies exactly the very same space in BC or BD that the whole occupied in AB under the ordinary atmospheric pressure of 30 inches, it follows that the cavity CD = BD — BC is equal to the bulk of the solid matter in the sand or powder. If we now find the number of grains of water held by the part CD of the tube, we determine at once the quantity of water equal in bulk to the solid matter in the sand or powder; and by comparing this with the weight of the sand, we obtain its exact specific gravity.

Brewster's Staktometer.

125. The Staktometer, or drop-measurer, is shown in fig. 34, where ABC is a glass vessel four or five inches long, having a hollow bulb B about half an inch in diameter. The instrument is filled by suction, and the fluid is discharged at C till it stands nearly at the point m, the zero of the scale. The fluid is then allowed to discharge itself at C by drops, and the number of them is counted till the surface of the fluid descends to another fixed point n. The experiment is then carefully repeated at different temperatures, till the number of drops of distilled water occupied by the cavity between m and n is accurately determined for various temperatures. The same experiment is made with alcohol. Thus, if N is the number of drops of distilled water whose specific gravity is S_p, and n the number of drops of alcohol whose specific gravity is s_a, and d the number of drops of any other mixture of alcohol and water contained in the same cavity m n, we shall have n — N:

\[ S - s = d - N : \frac{(d - N)(S - s)}{n - N} \]

and therefore \( S - \frac{(d - N)(S - s)}{n - N} \) will be the specific gravity of the mixture required.

With a small instrument, the number of drops of water between m and n was 724, whereas the number of drops of ordinary proof spirits was 2117 at 60° Fah. Now, as the specific gravity of the spirits was .920, and that of wa-

ter 1.000, we have a scale of 1393 drops for measuring all of specific specific gravities between .920 and 1.000, an unit in the Gravities fourth place of decimals corresponding to a variation of about two drops. From this experiment it follows that the bulk of a drop of water will be about 2.93 times as large as the bulk of a drop of the spirits.

Sike's Hydrometer.

126. This instrument, which is used in the collection of Sike's hydrometer, the revenue of the United Kingdom, is shewn in the annexed figure, where AB is a flat stem 3½ inches long, divided on each side into eleven parts, each of which is divided into two. This stem carries a brass ball BC, into which is fixed the conical stem CD, terminating in a loaded bulb DE. Eight circular weights, numbered as in the figure, can be placed on the conical stem CD. The square weight can be placed on the top of the stem. When the strength of spirits is to be measured, a weight is to be placed on CD capable of sinking the ball BC till the fluid surface cuts the stem AB. The number at the place where the stem is cut by the fluid, as seen from below, is then added to the number on the weight employed; and with this sum at the side, and the temperature of the spirits at the top, the strength per cent. is found in a table which accompanies the instrument.

The square weight shews the difference between the weight of proof spirit and that of water, as described in the first clause of the hydrometer act; and it is exactly one-twelfth part of the total weight of the hydrometer and weight 60. When this square weight is placed on the summit of the stem at A, and the instrument loaded with the weight No. 6, it will sink in distilled water at the temperature 51° to the proof point P, at that temperature, as indicated on the narrow edge of the stem.

SECT. III. On Tables of Specific Gravities.

127. As the knowledge of the specific gravities of bodies is of great use in all the branches of mechanical philosophy, specific we have given the following table, comprehending the gravities greater part of Brisson's tables, and one of the most extensive that has yet been published. When the specific gravities of any substance, as determined by different authors, seem to be at variance, the different results are frequently given, and the names of the observers prefixed by whom these results were obtained. The substances in the table have, contrary to the usual practice, been disposed in an alphabetical order. This was deemed more convenient for the purposes of reference, than if they had been divided into classes, or arranged according to the order of their densities.

The specific gravities of newly discovered minerals have been collected and inserted. The numbers are given in relation to water whose specific gravity is 1.000, excepting in the case of the gases, whose specific gravities are given in relation to that of atmospheric air, which is taken at 1.00. ## TABLE OF SPECIFIC GRAVITIES.

| Substance | Spec. Grav. | Alcohol, 13 parts, Water 3 parts, | Spec. Grav. | |----------------------------------|-------------|----------------------------------|-------------| | Of Specific Acacia, inspissated juice of, | 1.5153 | 12 | 0.8815 | | Gravities. Acid, nitric, | 1.2715 | 11 | 0.8947 | | nitric, highly concentrated, | 1.583 | 10 | 0.9075 | | muriatic, | 1.2847 | 9 | 0.9199 | | red acetous, | 1.0251 | 8 | 0.9317 | | white acetous, | 1.0135 | 7 | 0.9427 | | distilled acetous, | 1.0095 | 6 | 0.9519 | | acetic, | 1.007 | 5 | 0.9594 | | sulphuric, | 1.8409 | 4 | 0.9674 | | highly concentrated, | 2.125 | 3 | 0.9733 | | fluoric, | 1.500 | 2 | 0.9791 | | phosphoric, liquid, | 1.417 | 1 | 0.9852 | | solid | 2.852 | | 0.9919 | | citric, | 1.0345 | | Muschenbroek | | arsenic, | 3.391 | | Jardine | | of oranges, | 1.0176 | | 3.665 | | of gooseberries | 1.0581 | | 1.3586 | | of grapes, | 1.0241 | | 1.379 | | selenic, temp. 329°, | 2.524 | | Stromeyer | | boracic, in scales, | 1.475 | | 1.889 | | do. melted, | 1.803 | | Alum | | molybdic, | 3.460 | | 1.75 | | benzoic, | 0.667 | | 1.88 | | formic, | 1.102 | | Alumine | | Actinite, | 1.113 | | sulphate of | | Actinolite, glassy, | 3.24 | | saturated | | Adularia. See Felspar. | 2.950 | | solution of | | Eschinite, | 3.903 | | temp. 42° | | Agalmatolite, | 5.14 | | Watson | | Agate, oriental; | 2.500 | | 1.033 | | onyx, | 0.5901 | | 2.69 to 2.74 | | speckled, | 2.6375 | | Amber | | cloudy, | 2.607 | | yellow | | stained, | 2.6253 | | transparent | | veined, | 2.6324 | | opaque | | Icelandic, | 2.6667 | | green | | of Havre, | 2.348 | | Ambergris | | jasper, | 2.5881 | | 0.7800 | | Mocha, | 2.6356 | | 0.9263 | | iridescent, | 2.5535 | | Amblygonite | | Air, atmospheric, | 29.75 | | Amethyst | | Barom. | 0.00122 | | common. See | | Thermom. | 29.85 | | Rock crystal | | Barom. | 0.0012308 | | Amianthus | | Thermom. | 54.5 | | long, | | Alabaster of Valencia, | 2.638 | | penetrated | | veined, | 2.691 | | with water | | of Piedmont, | 2.693 | | short, | | of Malta, | 2.690 | | penetrated | | yellow, | 2.699 | | with water | | Spanish saline, | 2.713 | | Amianthinite | | oriental white, | 2.730 | | from Raschau | | ditto, semi-transparent, | 2.762 | | Bayreuth | | stained brown, | 2.744 | | Ammonia | | of Malaga, pink, | 2.8761 | | liquid, | | of Dalias, | 2.6110 | | muriate of | | Albite, | 2.624 | | saturated | | Alcohol, absolute, | 0.791 | | solution of | | highly rectified, | 0.8293 | | temp. 42° | | commercial, | 0.8371 | | Watson | | 15 parts, Water 1 part, | 0.8527 | | Amphibole | | Alcohol, | 0.8674 | | See Hornblende basaltic. | | Apatite. See Phosphorite. | | | Analcime | | Aploite | | | Andalusite, | | | | | or hardspar | | | | | Anhydrite, | | | | | Muriacite, | | | | | Anime, | | | | | oriental, | | | | | accidental, | | | | | Anorthite, | | | | | Anthophyllite| | | | | Antimony, | | | | | glass of, | | | | | in a metallic state, fused, | | | | | native, | | | | | grey, | | | | | sulphur of, | | | | | ore, grey and foliated, | | | | | radiated, | | | | | red, | | | | | Apatite. | | | | | See Phosphorite. | | | | | Aploite |

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**Note:** The table provides specific gravities for various substances, including acids, minerals, and gases, along with their corresponding alcohol-water mixtures and other properties such as color and transparency. | Substance | Spec. Grav. | Source | |----------------------------------|-------------|--------------| | Apophyllite | | | | Apple-tree, wood of the | | | | Aquamarine | | | | Arcanite | | | | Areca, inspissated juice of | | | | Aretizite, or Wernerite | | | | Argillite, or slate clay | | | | Arnott | | | | Arragonite | | | | Arsenic bloom, Pharmacolite, | | | | fused | | | | native | | | | glass of (arsenic of the shops) | | | | Arsenical pyrites, or Mispickel, | | | | See Realgar | | | | Asbestinite | | | | Asbestos, mountain cork | | | | penetrated with water | | | | ripe | | | | starry | | | | unripe | | | | Ash trunk | | | | dry | | | | Asphaltum, cohesive | | | | compact | | | | Assafetida | | | | Aventurine, semitransparent | | | | opaque | | | | Augite, or Pyroxene | | | | Automalite, Gahnite, or Fahlunite| | | | Axinite, or Thumerstone | | | | Azure stone, or lapis lazuli | | | | oriental | | | | of Siberia | | | | Barolite, or Witherite | | | | Barytes, or Baroselenite | | | | white | | | | grey | | | | rhomboidal | | | | octahedral | | | | in stalactites | | | | sulphate of, native | | | | carbonate of, native | | | | Baryto-calcite | | | | Basalt | | | | Basalt, from the Giant's Causeway| | | | prismatic, from Auvergne | | | | Baras, a juice of the pine | | | | Bay-tree, Spanish | | | | Beech-wood | | | | Beer, red | | | | white | | | | Benzoin | | | | Beryl, oriental | | | | occidental | | | | or aquamarine | | | | schlorous, or shortlite | | | | Bezoar, oriental | | | | occidental | | | | Bismuth, native | | | | sulphuretted | | | | ochre | | | | in a metallic state, fused | | | | Bismuth | | | | Bitumen of Judea | | | | Black-coal, pitch-coal | | | | slate-coal, English | | | | Bielschowitz | | | | cannel coal | | | | Blende, yellow | | | | brown, foliated | | | | black | | | | auriferous from Nagyag | | | | Blood, human | | | | crassamentum of serum of | | | | Blood-Stone | | | | Boles | | | | Bone of an ox | | | | Boracite | | | | Borax | | | | saturated solution of, temp. 42° | | | | Watson | | | | Bournonite | | | | Boxwood, French | | | | Dutch | | | | dry | | | | Brass common, cast | | | | wiredrawn | | | | cast, not hammered | | | | Brazil wood, red | | | | Brewsterite | | | | Bronzite | | | | Brick | | | | Bromine | | | | Bustamite | | | | Butter | | | | Cacao butter | | | | Cachibou, gum | | | | Cadmium (metal) not crushed | | | | crushed | | | | Calamine | | | | Substance | Spec. Grav. | |------------------------------------------------|------------| | Calamine | 4.100 | | Calcite | 1.700 | | Calculi, urinary | 1.240 | | Campeachy wood, or logwood | 0.9130 | | Camphor | 0.9887 | | Caoutchouc, elastic gum, or India rubber | 0.9335 | | Caragna, resin of the Mexican tree caragna | 1.1244 | | Carbon of compact earth | 1.3292 | | Carnelian, stalactite | 2.5977 | | speckled | 2.6137 | | veined | 2.6234 | | onyx | 2.6227 | | pale | 2.6301 | | pointed | 2.6120 | | arborized | 2.6133 | | Cat's eye | 2.500 | | grey | 2.625 | | yellow | 2.5675 | | blackish | 2.5573 | | Catchew, juice of an Indian tree | 1.3980 | | Caustic ammonia, solution of, or fluid volatile alkali | 0.897 | | Cedar tree, American | 0.5608 | | wild | 0.5608 | | Palestine | 0.5960 | | Indian | 1.3150 | | Celestine. See Strontian, sulphate of | | | Cerite | 4.500 | | Ceylanite, or Pleonaste | 3.765 | | Chabasie | 3.793 | | Chalcedony, bluish | 2.5867 | | onyx | 2.6151 | | veined | 3.6059 | | transparent | 2.6640 | | reddish | 2.6645 | | common | 2.600 | | Chalk | 2.655 | | Cherry-tree | 2.657 | | Chiastolite. See Macle | | | Chlorite | 2.775 | | Chloropal | 2.000 | | Chrysoberyl. See Cymophane | | | Chrysolite of the jewellers | 2.782 | | of Brazil | 2.692 | | Werner | 3.340 | | Chrysoprase, a variety of Chalcedony | 2.410 | | Crystal. See Rock Crystal | | | Crystalline lens | 1.100 | | Cinolite | 2.0 | | Cinnabar, dark red, from Deux-Ponts. Kirwan | 7.786 | | from Almaden, Brisson | 6.902 | | crystallized | 10.218 | | hepatic | 7.1 | | Cinnamon, volatile oil of | 1.044 | | Cinnamon-stone | 2.6 | | Citron-tree | 0.7263 | | Clinkstone | 2.575 | | Cloves, volatile oil of | 1.036 | | Cobalt, in a metallic state, fused | 7.645 | | Cobalt ore, grey | | | earthy, black, indurated | | | vitreous oxide of | | | Cocoa wood | | | Coccolite | | | Columbium | | | Condrodite | | | Copal, opaque | | | transparent | | | Madagascar | | | Chinese | | | Copper, native | | | from Siberia | | | Hungary | | | ore, compact vitreous | | | Cornish | | | purple, from Bannat | | | from Lorraine | | | Kirwan | | | Wiedemann | | | glance | | | pyrites | | | ore, white | | | grey | | | yellow | | | blue | | | foliated, florid, red | | | azure, radiated | | | emerald | | | muriate of | | | arseniate of | | | prismatic | | | partial arseniate | | | sulphate of, crystallised | | | saturated solution of sulphate of | | | drawn into wire | | | fished | | | Copper-sand, muriate of copper | | | Cork | | | Corundum of India | | | Cross stone. See Harmotome | | | Cryolite | | | Cube iron-ore | | | spar | | | Cubizite. See Analcite | | | Cyanite, Sappare, or Disthene | | | Cyder | | | Cymophane, or Chrysoberyl | | | Cypress wood, Spanish | |

| Substance | Spec. Grav. | |------------------------------------------------|------------| | Haüy | 5.511 | | Kirwan | 5.309 | | Gellert | 2.019 | | Muschenbroek | 2.425 | | Muschenbroek | 2.4405 | | Muschenbroek | 1.0403 | | Dandradra | 3.316 | | Hatchet | 5.918 | | 3.14 to 3.19 | | | 1.1398 | | | 1.0452 | | | 1.0600 | | | 1.0628 | | | Kirwan | 7.500 | | Haüy | 7.800 | | Gellert | 8.3084 | | Kirwan | 7.728 | | Kirwan | 4.129 | | Kirwan | 5.452 | | Kirwan | 4.956 | | La Metherie | 4.300 | | Kirwan | 4.983 | | Wiedemann | 5.467 | | Kirwan | 5.6 | | Kirwan | 4.080 | | Brisson | 4.344 | | La Metherie | 4.500 | | Haüy | 4.865 | | Kirwan | 4.5 | | Kirwan | 4.3 | | Kirwan | 3.2 | | Kirwan | 3.4 | | Wiedemann | 3.950 | | Wiedemann | 3.231 | | Brisson | 3.508 | | La Metherie | 2.850 | | Haüy | 3.300 | | Kirwan | 4.0 | | Kirwan | 4.3 | | hexahedral | 2.549 | | octahedral | 2.88 | | trihedral | 4.2 | | prismatic | 4.2 | | partial arseniate | 3.4 | | sulphate of, crystallised | 2.3438 | | saturated solution of sulphate of | 1.150 | | drawn into wire | 8.878 | | fished | 7.788 | | Hatchet | 8.985 | | La Metherie | 3.750 | | Herrgen | 4.431 | | Muschenbroek | 0.2400 | | Klaproth | 3.710 | | Bournon | 3.875 | | 3.981 | | | Cross stone. See Harmotome | | | Cryolite | 2.963 | | Cube iron-ore | 3.000 | | spar | | | Haüy | 2.964 | | Cyanite, Sappare, or Disthene | | | Saussure jun. Hermann | 3.517 | | 3.622 | | | Cyder | 1.0181 | | Cymophane, or Chrysoberyl | 3.600 | | Werner | 3.720 | | Haüy | 3.796 | | Muschenbroek | 0.6440 | | Substance | Specific Gravity | |-----------------------------------------------|------------------| | D | | | Datholite, from Arendal | 2.989 | | Dipyre, | 2.63 | | Diallage. See Smaragdite | 2.84 | | Diamond, oriental, colourless | 3.5212 | | rose-coloured, | 3.5310 | | orange-coloured, | 3.5500 | | green-coloured, | 3.5238 | | blue-coloured, | 3.5254 | | Diamond, Brazilian, yellow | 3.4444 | | orange, | 3.5185 | | Dichroite. See Iolite. | 3.55 | | Disthene. See Cyanite. | | | Dolomite, | 2.859 | | Dragon's blood, | 1.2045 | | E | | | Ebony, Indian, | Muschenbroek | | American, | 1.2090 | | Edingtonite, | Muschenbroek | | Elder tree, | 2.710 | | Eleni, | Muschenbroek | | 0.6950 | | | Elm trunk, | Muschenbroek | | 0.6710 | | | Emerald, | Gahn and Berzelius | | Werner, | 2.600 | | of Brazil, pseudo, | Gahn and Berzelius | | Epidote. See Zoisite. | Rose | | Epistiblite, | 2.749 | | Ether, sulphuric, | from 0.716 | | nitric, | to 0.745 | | muriatic, | 0.9088 | | acetic, | 0.7296 | | Euchroite, | 0.8664 | | Euclase, | 3.389 | | Eudialyte, | Haüy. | | Euphorbium gum, | 3.0625 | | F | | | Fahlunite. See Automalite. | | | Fat of beef, | 0.9232 | | veal, | 0.9342 | | mutton, | 0.9235 | | hogs, | 0.9368 | | Felspar, fresh, | Haüy. | | Adularia, | Struve | | Labrador stone, | Brisson | | glassy, | | | Fergusonite, | Borkowski | | Fettstein, | Haüy. | | Filbert tree, | Muschenbroek | | Fir, male, | Muschenbroek | | female, | Muschenbroek | | Fish-eye stone, Ichthyophalmite, or Apophyllite | Haüy. | | Flint, | Blumenbach | | olive, | 2.6057 |

1 The specific gravities of the gases are taken from Biot's Traité de Physique, tom. i. p. 383; from Gay Lussac's Tables in the Annales de Chimie et de Physique, vol. i. p. 218; and from Thomson's Annals of Philosophy, vol. i. p. 118. The measures for the gases, taken by MM. Biot and Arago, are calculated from Biot's formulae. They are given in relation to atmospheric air, which is supposed to be unity. | Substance | Spec. Grav. | |---------------------------------|-------------| | Gas, oxygen, mean | 1.103 | | nitrous gas, or deutoxide of azote, Bérard. | 1.0388 | | olefiant gas, Theodore, Saussure. | 0.97804 | | azote, Biot and Arago. | 0.96913 | | carbonic oxide, Cruickshank. | 0.9569 | | hydrocyanic vapour, Gay Lussac. | 0.9476 | | phosphuretted hydrogen, Sir H. Davy. | 0.870 | | steam, Tralles. | 0.6896 | | ammoniacal, Sir H. Davy. | 0.590 | | carburetted hydrogen, Thomson. | 0.555 | | Sir H. Davy. | 0.491 | | Cruickshank. | 0.678 | | Dalton. | 0.600 | | arsenical hydrogen, Trommsdorf. | 0.529 | | phosphuretted hydrogen, Dalton. | 0.852 | | Sir H. Davy. | 0.435 | | Thomson. | 0.073 | | hydrogen, Sir H. Davy. | 0.074 | | Biot and Arago. | 0.072098 | | Berzelius and Dulong. | 0.6885 | | Gay Lussite, Fuchs. | 2.78 | | Gehlenite, | 2.832 | | Gieseckite, | 4.000 | | Girasol, Brisson. | 1.300 | | Glance-coal, slaty, La Métherie.| 1.530 | | Kloproth. | 2.487 | | Glass, crown of St Louis, Cauchois, Biot. | 3.20 | | flint of M. Dartigues, Cauchois, Biot. | 3.192 | | flint used by Mr Tully for his achromatic telescopes. | 3.334 | | white flint, | 3.354 | | crown, | 3.437 | | common plate, | 2.520 | | yellow plate, | 2.760 | | white or French crystal, | 2.8922 | | St Gobins, | 2.4882 | | gall, | 2.8548 | | bottle, | 2.7325 | | Leith crystal, | 3.189 | | green, | 2.6423 | | borax, | 2.6070 | | fluid, | 3.329 | | of Bohemia, | 2.3959 | | of Cherbourg, | 2.5596 | | of St Cloud, animal, | 2.5459 | | mineral, | 2.2694 | | Glauberite, | 2.73 to 2.80| | Glaucina, | 3.000 | | Gmelinite, | 2.5 to 2.1 | | Gold, native, | 17.00 | | pure, of 24 carats, fine, fused, but not hammered, Hally. | 19.2587 | | the same hammered, | 19.342 | | English standard, 22 carats, fine, fused, but not hammered, | 18.888 | | guinea of George II. | 17.150 | | guinea of George III. | 17.629 | | Parisian standard, 22 carats, not hammered, | 17.486 | | Gold, the same hammered, | 15.709 | | Spanish gold coin, | 15.775 | | Holland ducats, | 17.9664 | | trinket standard, 20 carats, not hammered, | 17.4022 | | the same hammered, | 17.6474 | | Portuguese coin, | 17.5531 | | French money, 21½ carats, fused, coined, | 17.5531 | | French, in the reign of Louis XIII. | 2.6541 | | Granite, red Egyptian, | 2.7279 | | grey Egyptian, | 2.7609 | | beautiful red, | 2.7163 | | of Girardinor, | 2.6852 | | violet of Gyrogmagny, | 2.6431 | | red of Dauphiny, | 2.6836 | | green of Dauphiny, | 2.6678 | | radiated of Dauphiny, | 2.6384 | | red of Semur, | 2.7378 | | grey of Bretagne, | 2.6136 | | yellowish, of Carinthia, blue, Kirwan. | 2.9564 | | Granitelle, | 3.0526 | | of Dauphiny, | 2.8465 | | Graphic ore, | 1.4523 | | Graphite. See Plumbago. | 1.3161 | | Grenatite. See Staurolite. | 1.201 | | Gum Arabic, | 1.4817 | | tragacanth, | 1.4346 | | seraphic, cherry-tree, | 1.4456 | | Bassora, | 1.4206 | | Acajou, | 1.2216 | | Monbain, | 1.2071 | | Gutte, | 1.2289 | | ammoniac, | 1.196 | | Gayac, | 1.1390 | | liquid, from Botany Bay, lac, | 1.0284 | | animé, Eastern, | 1.0426 | | Western, | 0.836 | | Gunpowder in a loose heap, | 0.932 | | shaken, solid, | 1.745 | | Gypsum, opaque, | 2.1679 | | compact, specimen in the Leskean collection, | 2.939 | | compact, | 1.872 | | impure, foliated, mixed with granular limestone, Kirwan. | 2.725 | | semitransparent, | 2.3062 | | fine ditto, | 2.2741 | | opaque, rhomboidal, | 2.2642 | | ditto, 10 faces, | 2.3114 | | cuneiform, crystallised, | 2.3117 | | striated of France, | 2.3057 | | of China, | 2.3088 | | flowered, | 2.3059 | | sparry opaque, | 2.2746 | | semitransparent, | 3.3108 | | Gypsum, granularly foliated, in the Leskean collection, Kirwan. | 2.900 | | mixed with marl, of a slaty form, | 2.473 | | Harmotome, or Cross-Stone, Hazel, Muschenbroek. | 2.3333 | | Hazel, | 0.606 | | Specie | Spec. Grav. | |--------|------------| | Gmelin | 2.687 | | Gimonde | 3.333 | | Kirwan | 2.629 | | Blumenbach | 2.633 | | Hematites. See Ironstone. | 2.11 | | Herschelite | 2.944 | | Hollow spar, Chalcolite | 2.8763 | | Hone, razor, white, penetrated with water, razor, white and black | 2.8839 | | Honey | 1.4500 | | Honeystone, or Mellite | 1.586 | | Hopeite | 2.61 | | Hornblende, common | 3.600 | | Schiller spar | 3.830 | | schistose | 2.882 | | basaltic | 2.909 | | Reus | 3.155 | | Kirwan | 3.150 | | Kirwan | 3.220 | | Kirwan | 3.333 | | Hornstone, or petrosilix, ferruginous, veined, grey, blackish-grey, yellowish-white, bluish, and partly yellowish-grey, dark purplish-red iron-shot, greenish-white with reddish spots, from Lorraine, iron shot, brownish-red, outside bluish, grey inside | 2.530 | | Humboldtite | 2.813 | | Hyalite | 2.110 | | Hyacinth | 4.000 | | Klaproth | 4.545 | | Hydargillite. See Wavellite. | 4.620 | | Hydrogen, bicarburet of, at 60° | 0.85 | | Hyperstene. See Bronzite. | | | Hypocist | 1.5263 | | Hyposulphite of lime | 1.0105 | | Jade, or Nephrite, white, green, olive, from the East Indies, of Switzerland, combined with the boracic acid and boric cited calx | 2.9592 | | Jasmin, Spanish | 2.9660 | | Jasper, veined, red, brown, yellow, violet, grey, cloudy, green, bright green, deep green, brownish-green, blackish, blood coloured | 2.6277 | | Jasper, onyx, flowered, red and white, red and yellow, green and yellow, red, green, and grey, red, green, and yellow, universal, agate | 2.8160 | | Idocrase. See Vesuvian. | | | Jenite | 3.80 | | Jet, a bituminous substance | 4.00 | | Indigo | 1.2590 | | Indigo, penetrated with water | 0.7690 | | Insipidated juice of liquorice | 1.0095 | | Iodine | 1.7228 | | Thomson | 3.0844 | | Gay Lussac | 4.948 | | Iolite, or Dichroite | 2.56 | | Iridium. See Osmium. | | | fused by galvanism | 18.68 | | Iron, native, meteoric, chromate of, from the department of Var, from the Uralian mountains, in Siberia, sulphate of, crystallized, saturated solution, temp. 42° | 6.48 | | Laugier | 4.0326 | | Ure | 1.7774 | | arseniate of, fused, but not hammered | 3.000 | | forged into bars | 7.200 | | pyrites, dodecahedral, from Freyberg, Cornwall, cubic | 7.600 | | Hatchet | 7.788 | | Gellert | 4.830 | | Kirwan | 4.682 | | Brisson | 4.789 | | Hatchet | 4.702 | | Hatchet | 4.698 | | Hatchet | 4.775 | | Hailey | 4.518 | | magnetic, white, sand, magnetic sand, from Virginia | 4.600 | | Bergman | 7.800 | | magnetic | 4.200 | | magnetic | 4.900 | | ore specular | 4.793 | | Kirwan | 5.139 | | ore specular | 4.939 | | Brisson | 5.218 | | micaceous | 4.728 | | Kirwan | 5.070 | | Ironstone, red, ochry, compact, from Siberia, Lancashire, compact, brown, from Bayreuth, from Tyrol | 2.952 | | Kirwan | 3.423 | | Kirwan | 3.760 | | Brisson | 3.573 | | Wiedemann | 3.863 | | Kirwan | 3.551 | | Kirwan | 3.753 | | Brisson | 3.503 | | Kirwan | 3.477 | | Kirwan | 3.005 | | Gellert | 4.740 | | Kirwan | 3.951 | | Gellert | 3.789 | | Wiedemann | 4.029 | | Kirwan | 3.640 | | Kirwan | 3.810 | | Brisson | 3.672 | | Kirwan | 3.300 | | Kirwan | 3.600 | | Wiedemann | 4.076 | ### HYDRODYNAMICS

| Substance | Spec. Grav. | Author | |----------------------------------|-------------|--------------| | Ironstone, clay reddie | 3.139 | Brisson | | clay, lenticular | 2.931 | Blumenbach | | clay, common, from Cathina at Raschau | 2.673 | Kirwan | | from Roscommon in Ireland | 3.471 | Rotheram | | Carron in Scotland | 3.205 | Rotheram | | clay, reniform iron-ore | 2.574 | Wiedemann | | clay, pea-ore | 5.207 | Malinghof | | Iron, native (Heleachen mass) | 6.723 | Monheim | | ore, lowland, from Sprottau | 2.944 | Kirwan | | Iserine, an oxide of titanium from the Iser in Bohemia | 4.500 | Muschenbroek | | Juniper tree | 0.5560 | Muschenbroek| | Ivory, dry | 1.8250 | Muschenbroek| | Ivy gum, from the Hedera terrestris | 1.2948 | Muschenbroek | | Keffekil, or Meerschaum | 1.6000 | Klaproth | | Kinkina | 0.7840 | Muschenbroek| | Knebelite | 3.714 | Muschenbroek| | Kyanite. See Cyanite | | | | Labdanum, resin | 1.1862 | | | in tortis | 2.4933 | | | Lapis lazuli. See Azure stone | | | | Laumonite | 2.20 | | | Lard | 0.9478 | | | Latrobite | 2.720 | | | Latalite. See Havyme | | | | Lead-glance, or galena, common | 7.290 | Gellert | | from Derbyshire | 6.565 | Watson | | compact | 7.785 | | | crystallized, radiated | 6.886 | Gellert | | from the Hartz | 7.444 | Kirwan | | Kautenbach | 5.052 | | | Kirschwalder | 7.587 | Brisson | | ore, corneous | 5.500 | La Métherie | | Kirschwalder | 7.448 | Kirwan | | Kautenbach | 6.140 | Vauquelin | | Kirschwalder | 5.820 | Vauquelin | | Chenevix | 6.065 | | | Bindheim | 3.920 | | | of black lead | 6.745 | | | blue | 5.461 | Gellert | | brown | 6.974 | Wiedemann | | from Huguelgoet | 6.600 | Klaproth | | black | 6.909 | Haüy | | white, from Leadhills | 5.770 | Gellert | | phosphorated, from Wanlockhead | 7.236 | Chenevix | | black | 6.559 | Haüy | | selenated, | 6.560 | Klaproth | | Zschoppau | 7.697 | | | Brisgaw | 6.270 | Klaproth | | red, or red lead spar | 6.941 | Haüy | | sulphato-carbonate | 6.027 | Gellert | | cupreous do | 6.8 to 7.0 | | | Lead | 11.352 | Fischer, Wollaston | | arseniate of | 11.445 | Gellert | | carbonat e of | 5.00 | | | | 6.40 | | | | 6.00 | | | | 7.20 | |

| Substance | Spec. Grav. | Author | |----------------------------------|-------------|--------------| | Lead, muriate of | 7.07 | | | murio-carbonate of | 6.00 to 6.1 | | | sulphate of | 6.3 | | | chromate of | 6.00 | | | acetate of | 2.3953 | Muschenbroek | | vitriol, from Anglesea | 6.300 | Klaproth | | Lepidolite, lilalite | 0.7033 | Muschenbroek | | Lemon tree | 2.816 | Klaproth | | Leucolite. See Dipyre | 2.854 | Haüy | | Leucite, or Amphigene | 2.455 | Klaproth | | Lignum vitae | 1.3330 | Muschenbroek | | Limestone, compact | 2.7200 | | | foliated | 2.710 | | | granular | 2.837 | | | green, arenaceous | 2.700 | | | Linden wood | 2.800 | | | Lithomarge | 3.183 | | | Logwood, or Campeachy wood | 2.742 | | | Macle, or chiastolite | 0.604 | | | Madder root | 2.50 | | | Mahogany | 1.232 | Watson | | Magnesia, sulphate of, crystallized use, Ure | 2.350 | | | saturated solution, temp. 42° | 1.7976 | | | native, hydrate of | 2.200 | | | Magnesite, or carbonate of magnesia, a new species, from Baumgarten in Silesia | 2.95 | Hausmann | | Magnetic pyrites. See Iron | 3.572 | Brisson | | Malachite, compact | 3.641 | | | Manganese | 3.994 | Muschenbroek | | grey ore of, striated | 6.850 | Bergman | | grey, foliated | 7.000 | Hielm | | grey, foliated, red, from Kapnick | 4.249 | Brisson | | black | 4.756 | Rimann | | black | 4.181 | | | black | 3.742 | Hagen | | black | 3.233 | Kirwan | | black | 2.0000 | Dolomieu | | black | 3.0000 | | | black | 3.7076 | Brisson | | scaly | 4.1165 | | | sulphuret of | 3.95 | | | white | 2.8 | | | phosphate of | 3.439 | Vauquelin | | Maple wood | 3.775 | Ullman | | Marble Carrara | 0.7550 | Muschenbroek| | Pyrenean | 2.716 | Brisson | | black Biscayan | 2.726 | | | Brocatelle | 2.695 | | | Castilian | 2.650 | | | Valencian | 2.700 | | | Grenadian white | 2.710 | | | Siennian | 2.705 | | | Roman violet | 2.678 | | | African | 2.755 | | | Italian, violet | 2.708 | | | Norwegian | 2.858 | | | Siberian | 2.728 | | | French | 2.649 | | | Substance | Spec. Grav. | |------------------------------------------------|-------------| | Marble, Switzerland | 2.714 | | Egyptian, green | 2.668 | | yellow, of Florence | 2.516 | | Marmolite | 2.470 | | Mastic | 1.0742 | | tree | Muschenbroek| 0.8490 | | Medlar tree | Muschenbroek| 0.9440 | | Meerschaum. See Kesekil | | | Meiomite | Karsten | 3.691 | | Melanite, or black garnet | Werner | 3.800 | | Memachanite | Lampadius | 4.270 | | Mercurial hepatic ore, compact | Gregor | 4.227 | | Mercury at 32° of heat | Kirwan | 7.186 | | at 60° | Gellert | 7.352 | | at 62° | Faraday | 13.568 | | at 212° | | | | at 3°-42 centigrade | Fischer | 13.58597 | | in a solid state, 40° below 0° Fahr | Biddle | 15.612 | | in a fluid state, 47° above 0 | Biddle | 13.545 | | native | Haüy | 13.5681 | | corrosive muriate of | | 6.49 | | saturated solution, temp. 42°, Watson | | 1.037 | | natural calx of | | 9.230 | | precipitate, per se | | 10.871 | | red | | 8.359 | | mineralized by sulphur, native Ethiops | Hahn | 2.233 | | See also Cinnabar | | | | Mesotype | | 2.0833 | | Mica, biaxal | | 2.883 | | Haüy | | 2.6546 | | Milk, woman's | | 1.0203 | | mare's | | 1.0346 | | ass's | | 1.0355 | | goat's | | 1.0341 | | ewe's | | 1.0409 | | cow's | | 1.0324 | | Mineral pitch, elastic, or asphaltum | Hatchet | 0.905 | | La Métherie | | 1.233 | | tallow | | 0.930 | | Molybdate of lead | | 0.770 | | Molybdena in a metallic state, saturated with water | Schumacher | 4.667 | | native | Brisson | 4.7385 | | Mountain Crystal. See Rock-Crystal | | | | Mulberry tree, Spanish | Muschenbroek| 0.8970 | | Muriacite. See Anhydrite | | | | Muricalcite, crystallized, or rhomb spar | | 2.480 | | Myrrh | | 1.3600 | | Natrolite Swedish | Thomson | 2.773 | | red crystals | | 2.790 | | Naphtha, liquid | | 2.168 | | Naphthaline | | 0.8475 | | Nepheline, or Sommite | Haüy | 3.2741 | | Neprhite. See Jude | | | | Nickel in a metallic state | Bergmann | 7.421 | | copper | Brisson | 9.3333 | | Nickel copper | Gellert | 7.560 | | Nickel, ore of, called arsenical nickel, or Kupfernickel of Saxony | | 6.648 | | Kupfernickel of Bohemia | | 6.607 | | sulphuretted, forged | Richter | 6.620 | | and antimony, sulphuret of | | 6.451 | | Nickeline, a metal discovered by Richter, cast | Richter | 8.55 | | Nigrine, or calcareo-siliceous titanic ore | Vaquelin | 3.700 | | Klapproth | | 4.445 | | Louwitz | | 4.673 | | Nitre | | 1.9000 | | crystallized, quadrangular | Ure | 2.0060 | | saturated solution of, temperature 42° | Watson | 1.095 | | Novaculite, or Turkey hone. See Slate Whet. | | | | Oak, 60 years old, heart of | Muschenbroek| 1.1700 | | Obsidian | | 2.348 | | Octohedrite | Haüy | 3.857 | | Oil of filberts | | 0.916 | | walnut | | 0.92 | | hemp-seed | | 0.9258 | | poppies | | 0.9238 | | rape-seed | | 0.9193 | | lint-seed | | 0.9403 | | poppy-seed | | 0.929 | | whale | | 0.9233 | | ben, a tree in Arabia | | 0.9119 | | beechnast | | 0.9176 | | codfish | | 0.9233 | | olives | | 0.9153 | | almonds, sweet | | 0.9170 | | volatile of mint, common | | 0.8982 | | sage | | 0.9016 | | thyme | | 0.9023 | | rosemary | | 0.9057 | | calamint | | 0.9116 | | cochlearia | | 0.9427 | | wormwood | | 0.9073 | | tansy | | 0.9328 | | Stragan | | 0.9949 | | Roman camomile | | 0.8943 | | sabine | | 0.9294 | | fennel | | 0.9294 | | fennel-seed | | 1.0083 | | coriander-seed | | 0.8655 | | caraway-seed | | 0.9049 | | dill-seed | | 0.9128 | | anise-seed | | 0.9867 | | juniper-seed | | 0.8577 | | cloves | | 1.0363 | | cinnamon | | 1.0439 | | turpentine | | 0.8697 | | amber | | 0.8865 | | the flowers of orange | | 0.8798 | | lavender | | 0.8938 | | myssop | | 0.8892 | | Olibanum gum | | 1.1732 | | Olive tree | | 0.9072 | | copper ore, foliated | Bournon | 4.281 | | fibrous | Bournon | 4.281 | | Olivine. See Peridot | | | | Opal, precious | Blumenbach | 2.114 | | common | Klaproth | 1.958 | | common | | 2.015 | | Substance | Spec. Grav. | |------------------------------------------------|------------| | Opal, common, semi-opal reddish, from Telkobanya | 2.144 | | ligniform, or wood | 2.600 | | Opium | 13.365 | | Ophites. See Porphyry Hornblende | | | Opoponax | 1.6226 | | Orange tree | 0.7059 | | Orpiment | 3.048 | | Orpiment, red. See Realgar | 3.435 | | Osmium and Iridium, alloy of | 19.5 | | Palladium | 11.8 | | Paranthine. See Scapolite | | | Pear-tree | 0.6610 | | Pearl-stone | 2.34 | | Pearls, oriental | 2.683 | | Peat, hard | 1.329 | | Peridot, or Olivine | 3.428 | | Peruvian bark | 3.225 | | Petalite | 2.440 | | Petroleum | 0.8783 | | Petrosilex. See Hornstone | | | Pharmacolite, or Arseniate of Lime | 2.640 | | Phosphorite, or Spargel stone, whitish, from | 2.8249 | | Spain, before absorbing water | 2.8648 | | after absorbing water | 3.098 | | greenish, from Spain | 3.218 | | Saxon | 1.770 | | Phosphorus | 2.320 | | Pierre de volvic | | | Pinite | 2.980 | | Pitch ore, or sulphuretted uranite | 6.378 | | Pitch-stone, black | 5.530 | | yellow | 2.0850 | | red | 2.6695 | | brick red, from Misnia | 2.720 | | leek green, inclining to olive | 2.298 | | pearl gray | 1.970 | | blackish | 2.3191 | | olive | 2.3145 | | dark green | 2.3149 | | Pithy iron-ore | 3.956 | | Plasma | 2.04 | | Platina | 20.722 | | drawn into wire | 21.0417 | | a wedge-of, sent by Admiral Gravina to Mr Kirwan| 20.663 | | a bar of, sent by the King of Spain to the King of Poland | 20.722 | | in grains, purified by boiling in nitrous acid | 17.500 | | native | 15.601 | | fused | 17.200 | | purified and forged | 14.626 | | milled and purified | 20.336 | | compressed by a flating mill | 20.98 | | Pleonaste. See Ceylonite | | | Plum tree | 0.7850 | | Plumbago, or graphite | 1.987 | | Polymignite | | | Pomegranate tree | | | Poplar wood | | | white Spanish | | | Porcelain from China | | | Seves, hard | | | tender | | | Saxony, modern | | | Limoges | | | of Vienna | | | Saxony, called Petite Jaune | | | Porcellanite | | | Porphyry, green | | | red | | | red of Dauphiny | | | red from Cordova | | | green from ditto | | | hornblende, or orphites | | | Pitch-stone | | | mullen | | | sand-stone | | | Potash, carbonate of | | | bicarbonate of | | | fixed subcarbonate of | | | muriate of | | | tartrate of, acidulous | | | antimonial | | | sulphate of | | | Potassium at 15° centigrade | | | Proof spirit, according to the English excise laws | 0.916 | | Pumice-stone | | | Pycnite, or shorlous beryl | | | Pyrolite | | | Pyrites. See Copper and Iron | | | Pyrope | | | Pyrochlore | | | Pyrophysalite | | | Pyrorhite | | | Pyroxene. See Augite | | | Quartz, crystallized, brown, red | | | brittle | | | crystallized | | | milky | | | elastic | | | Realgar, or red orpiment | | | Resin, or Guiacum | | | Retinasphalt | | | Rhizitite | | | Rhodium | | | Rock-crystal, from Madagascar | | | clove brown | | | snow white from Marmerosch | | | crystal, European, pure, gelatinous | | | Rock-crystal | Spec. Grav. | |-------------|------------| | Malus | 2.63717 | | | 2.6526 | | | 2.6497 | | | 2.6701 | | | 2.6542 | | | 2.5818 | | | 2.6535 | | | 2.6570 | | | 2.6513 | | | 2.6534 | | | 2.6536 | | | 3.6096 | | | 0.5956 | | | 1.1450 | | | 4.2833 | | | 3.5311 | | | 3.7600 | | | 3.5700 | | | 3.6458 | | | 4.102 | | | 4.246 | | | 3.1 | | | 3.5 |

| Schorl, cruciform | Spec. Grav. | |------------------|------------| | violet of Dauphiny | 3.2861 | | green | 3.2956 | | common | 3.4529 | | Brisson | 3.092 | | Gerhard | 3.150 | | Kirwan | 3.212 | | Selenite, or broad foliated gypsum | 2.322 | | Serpentine, opaque, green, Italian, penetrated with water | 2.4295 | | ditto, red and black veined | 2.4729 | | ditto, veined; black and olive, semitransparent, grained | 2.6273 | | ditto, fibrous | 2.5939 | | ditto, from Dauphiny, opaque, spotted black and white, spotted black and grey, spotted red and yellow, green from Grenada, deep green from Grenada, black, from Dauphiny, or variolite, green from Dauphiny, green, yellow, violet, of Dauphiny | 2.5859 | | Shale | 2.6 | | Siderocalcite, or brown spar | 2.7913 | | Sidero-schisolite | 2.837 | | Silver, sulphuretted, or silver-glance | 3.00 | | La Métherie | 6.910 | | brittle | 7.200 | | white | 7.208 | | red, or ruby | 5.3 | | light red | 5.564 | | sooty | 5.5886 | | native, common | 5.443 | | antimonial | 5.592 | | auriferous | 10.000 | | ore, dark red | 10.333 | | ore, corneous, or horn ore | 9.4406 | | arseniated, ferruginous, penetrated with water | 10.000 | | ore | 4.7488 | | Gellert | 4.804 | | virgin, 12 denier, fine, not hammered | 10.474 | | 12 deniers, hammered | 10.510 | | Paris standard, 11 deniers | 10.175 | | 10 grains, fused, hammered | 10.376 | | shilling of George II | 10.784 | | George III | 10.534 | | French money, 10 deniers, 21 grains, fused | 10.048 | | French money, 10 deniers, 21 grains, coined | 10.408 | | Sinople, coarse jasper | 2.6913 | | Slate clay. See Argillite | 2.6718 | | common, or schistus, common | 2.6718 | | penetrated with water | 2.6905 | | whet, or novaculite | 0.722 | | Isabella yellow | 2.609 | | stone | 2.1861 | | fresh polished | 2.7664 | | Substance | Spec. Grav. | |------------------------------------------------|-------------| | Slate adhesive, new | 2.080 | | Siliceous, Kirwan | 2.596 | | Horn, or schistose porphyry, Kirwan | 2.512 | | Smalt, or blue glass of cobalt | 2.440 | | Smaragdite, from Corsica | 3.000 | | Soda, sulphate of muriate | 1.4398 | | Saturated solution, temperature 42°, Watson | 1.198 | | Nitrate of | 2.09 | | Carbonate of | 1.0 to 1.5 | | Sesquicarbonate of | 2.11 | | Tartrate of, saturated solution of, Watson | 1.114 | | Fossil | 2.1430 | | Saturated solution of, temperature 42°, Watson | 2.054 | | Sodalite, Sodium, at 15° centigrade, Gay-Lussac and Thenard | 0.86307 | | Somnite. See Nepheline | 2.530 | | Sordawalite | | | Spar, brown. See Sidero-Calcite | | | White sparkling | 2.5946 | | Red ditto | 2.4378 | | Green ditto | 2.7045 | | Blue ditto | 2.6925 | | Green and white ditto | 3.1051 | | Transparent ditto | 2.5644 | | Adamantine. See Corundum | 3.873 | | Schiller. See Hornblend Labrador | | | Fluor, white | 3.1555 | | Red, or false ruby | 3.1911 | | Octahedral | 3.1815 | | Yellow, or false topaz | 3.0967 | | Green, or false emerald | 3.1817 | | Octahedral | 3.1838 | | Blue, or false sapphire | 3.1688 | | Greenish blue, or false aquamarine | 3.1820 | | Violet, or false amethyst | 3.1757 | | English | 3.1796 | | Of Auvergne | 3.0943 | | In stalactites | 3.1668 | | Pearl, or bitter (carb. of lime and magnesia), calcareous rhomboidal | 2.8378 | | In tubes | 2.7151 | | Of France | 2.7140 | | Prismatic | 2.7182 | | And pyramidal | 2.7115 | | Pyramidal | 2.7141 | | Puant gris | 2.7121 | | Puant noir | 2.6207 | | Or flox ferri | 2.6747 | | Spargel stone. See Phosphorite | | | Spermaceti | 0.9433 | | Spinelle. See Ruby | | | Sphele. See Rutile | | | Spirit of wine. See Alcohol | | | Spodumene, or triphane | 3.1923 | | Dandrada | 3.218 | | Stalactite transparent, opaque | 2.3239 | | Penetrated with water | 2.4783 | | Staurolite, staurolite, or grenatite | 2.5462 | | Steatites of Bareight, penetrated with water | 3.286 | | Indurated | 2.6149 | | Tabasheer | | | Taeanahaca, resin | | | Tachylite | | | Talc, black crayon | | | Ditto German | |

| Substance | Spec. Grav. | |------------------------------------------------|-------------| | Steatites of Bareight, penetrated with water | 2.6757 | | Indurated | 2.5834 | | Muschenbroek | 2.6322 | | Jardine | 2.059 | | Tabasheer | 2.412 | | Taeanahaca, resin | 1.0463 | | Tachylite | 2.8534 | | Talc, black crayon | 2.52 | | Ditto German | 2.080 | | Ditto German | 2.246 | | Substance | Spec. Grav. | |----------------------------------|------------| | Talc, yellow | 2.655 | | white | 2.704 | | mercury | 2.7917 | | black | 2.9001 | | earthy | 2.6325 | | slaty | 2.718 | | common Venetian | 2.700 | | indurated | 2.800 | | Tallow | 2.90 | | Tantalite | 0.9419 | | Tantalum metal | 7.953 | | in large masses | 5.61 | | in small pieces | 6.291 | | Tartar | 6.208 | | Terra Japonica | 1.8490 | | Tellurium native | 1.3980 | | graphic | 5.7 | | yellow, of Naygag | 10.67 | | black | 8.9 | | Tennantite | 4.375 | | Themandite | 2.73 | | Thomsonite | 2.35 to 2.4| | Tin, pure, from Cornwall, fused | 7.170 | | fused and hammered | 7.291 | | of Malacca, fused | 7.296 | | fused and hammered | 7.306 | | of Galicia | 7.063 | | of Ehrenfriedsdorf in Saxony | 7.271 | | pyrites | 4.350 | | stone | 4.785 | | Gellert | 6.300 | | Brunnich | 6.989 | | black | 6.750 | | red | 6.901 | | red | 6.9348 | | fibrous | 5.845 | | Werner | 6.970 | | Blumenbach | 7.000 | | new, fused | 5.800 | | fused and hammered | 6.450 | | fine, fused | 7.3013 | | fused and hammered | 7.3115 | | common | 7.4789 | | called Claire-étoffe | 7.5194 | | ore, Cornish | 7.9200 | | ore | 8.4869 | | from Fahlun, Gahn and Berzelius | 5.800 | | stone, white | 6.008 | | Titanite, Rutlitie, or Sphene | 4.102 | | La Metherie | 4.246 | | Topaz, oriental | 4.0106 | | Brazilian | 3.5355 | | from Saxony | 3.5640 | | oriental pistachio | 4.0615 | | Saxony white | 3.5535 | | greenish blue | 3.5489 | | red | 3.5311 | | Tourmaline | 3.086 | | green | 3.362 | | blue | 3.155 | | Tremolite | 2.931 | | Triclasite | 2.61 to 2.66| | Torrelite | 3.25 | | Triphane, See Spodumene | |

| Substance | Spec. Grav. | |----------------------------------|------------| | Tungstate of lead | 8.1 | | Tungsten | 4.355 | | Leysser | 5.800 | | Kirwan | 6.028 | | Brisson | 6.066 | | Klaproth | 6.015 | | Turbeth mineral | 5.570 | | Turpentine, spirits of | 8.235 | | liquid | 0.870 | | Turquoise, ivory tinged by the | 2.500 | | blue calx of copper | 2.908 | | oriental, or calaite | 2.83 to 3.0| | Ultramarine | 2.360 | | Desormes and Clement | 2.19 | | Uran glimmer, or Uranite | 6.440 | | Uranite in a metallic state | 3.150 | | sulphuretted. See Pitch ore | 3.2438 | | Uranitic ochre indurated | 7.500 | | Urine, human | 3.19 | | diabetic | 1.015 | | Henry | 1.028 | | Vanadiate of lead | 1.040 | | Vauquelinite | 6.99 to 7.23| | Vermeille, a kind of oriental | 5.5 to 5.78| | ruby | 4.2299 | | Vesuvian | 3.575 | | Wiedemann | 3.420 | | of Siberia | 3.365 | | Klaproth | 3.339 | | Haugy | 3.407 | | Vine | 1.2370 | | Vinegar, red | 1.0251 | | white | 1.0135 | | radical | 1.080 | | Vitriol, Dantzic | 1.715 | | Walnut-tree of France | 0.6710 | | Wagnerte | 3.11 | | Levy | 3.01 | | Water distilled at 32° of Fahr | 1.0000 | | sea | 1.0263 | | of Dead Sea | 1.2403 | | Gay-Lussac | 1.2283 | | wells | 1.0017 | | of Bareges | 1.00037 | | of the Seine filtered | 1.00015 | | of Spa | 1.0009 | | of Armeil | 1.00046 | | Avray | 1.00043 | | Seltzer | 1.0305 | | Wavelite, or hydrargillite, from | 2.337 | | Barnstaple | 0.9648 | | Wax, bees | 0.9686 | | shoemakers | 0.897 | | Whey, cows | 1.019 | | Willow | 0.5850 | | Wine of Torrins, red | 0.9930 | | white | 0.9876 | | Champagne, white | 0.9979 | | Pakarate | 0.9997 | | Xeres | 0.9924 | | Malmsey Madeira | 1.0382 | CHAPTER III.—ON CAPILLARY ATTRACTION, AND THE COHESION OF FLUIDS.

128. We have already seen, when discussing the equilibrium of fluids, that when water or any other fluid is poured into a vessel, or any number of communicating vessels, its level in each vessel will be horizontal, or it will rise to the same height in each vessel, whatever be its form or position. This proposition, however, only holds true when the diameter of these vessels or tubes exceeds the fifteenth of an inch; for if a system of communicating vessels be composed of tubes of various diameters, the fluid will rise to a level surface in all the tubes which exceed one fifteenth of an inch in diameter; but in the tubes of a smaller bore, it will rise above that level to altitudes inversely proportional to the diameters of the tubes. The power by which the fluid is raised above its natural level is called capillary attraction, and the glass tubes which are employed to exhibit its phenomena are named capillary tubes. These appellations derive their origin from the Latin word capillus, signifying a hair, either because the bores of these tubes have the fineness of a hair, or because that substance is itself supposed to be of a tubular structure.

129. When we bring a piece of clean glass in contact with water or any other fluid, except mercury and fused metals, and withdraw it gently from its surface, a portion of the fluid will not only adhere to the glass, but a small force is necessary to detach this glass from the fluid mass, which seems to resist any separation of its parts. Hence it is obvious that there is an attraction of cohesion between glass and water, and that the constituent particles of water have also an attraction for each other. The suspension of a drop of water from the lower side of a plate of glass is a more palpable illustration of the first of these truths; and the following experiment will completely verify the second. Place two large drops of water on a smooth metallic surface, their distance being about the tenth of an inch. With the point of a pin unite these drops by two parallel canals, and the drops will instantly rush to each other through these canals, and fill the dry space that intervenes. This experiment is exhibited in fig. 36, where AB is the metallic plate, C, D the drops of water, and m, n, the two canals.

130. Upon these principles many attempts have been made to account for the elevation of water in capillary tubes; but most of the explanations which have hitherto been offered, are founded upon hypothesis, and are very far from being satisfactory. Without presuming to substitute a better explanation in the room of those which have been already given, and so frequently repeated, we shall endeavour to illustrate that explanation of the phenomena of capillary attraction which seems liable to the fewest objections. For this purpose let E be a drop of water laid upon a clean glass surface AB. Every particle of the glass immediately below the drop E, exerts an attractive force upon the particles of water. This force will produce the same effect upon the drop as a pressure in the opposite direction, the pressure of a column of air, for instance, on the upper surface of the drop. The effect of the attractive force, therefore, tending to press the drop to the glass will be an enlargement of its size, and the water will occupy the space FG; this increase of its dimensions will take place when the surface AB is held downwards; and that it does not arise from atmospheric pressure may be shewn by performing the experiment in vacuo. Now, let A B (fig. 37) be a section of the plate of glass, AB (fig. 36), held vertically, part of the water will descend by its gravity, and form a drop B, while a small film of the fluid will be supported at m by the attraction of the glass. Bring a similar plate of glass CD, into a position parallel to AB, and make them approach... apillary nearer and nearer each other. When the drops B and D come in contact, they will rush together from their mutual attraction, and will fill the space o p. The gravity of the drops B and D being thus diminished, the films of water at m and n, which were prevented from rising by their gravity, will move upwards. As the plates of glass continue to approximate, the space between them will fill with water, and the films at m and n being no longer prevented from yielding to the action of the glass immediately below them, (by the gravity of the water at o p, which is diminished by the mutual action of the fluid particles) will rise higher in proportion to the approach of the plates. Hence it may be easily understood how the water rises in capillary tubes, and how its altitude is inversely as their internal diameters. For let A, a be the altitudes of the fluid in two tubes of different diameters D, d; and let C, c be the two cylinders of fluid which are raised by virtue of the attraction of the glass. Now, as the force which raises the fluid must be as the number of attracting particles, that is, as the surface of the tube in contact with the water, that is, as the diameter of the tubes, and as this same force must be proportional to its effects on the cylinder of water raised, we shall have D : d = C : c.

But (Geometry, Sect. VIII. Theor. XI. Sect. IX. Theor. II.) C : c = D^2A : d^2a, therefore D^2A : d^2a = D : d; hence

\[ D^2A \cdot d = d^2a \cdot D, \text{ and } DA = \frac{d^2aD}{D^2a}, \text{ or } DA = da, \text{ that is, } D : d = a : A, \text{ or the altitudes of the water are inversely as the diameters of the tubes. Since } DA = da, \text{ the product of the diameter by the altitude of the water will always be a constant quantity. In a tube whose diameter is 0.01, or } \frac{1}{100} \text{ of an inch, the water has been found to reach the altitude of 5.3 inches; hence the constant quantity } 5.3 \times 0.01 = 0.053 \text{ may fitly represent the attraction of glass for water. According to the experiments of Muschenbroek, the constant quantity is 0.039; according to Weitbrecht 0.0428; according to Monge 0.042, and according to Atwood 0.0530. When a glass tube was immersed in melted lead, Gellert found the depression multiplied by the bore to be 0.0054.}

131. Having thus attempted to explain the causes of capillary action, we shall now proceed to consider some of its more interesting phenomena. In fig. 38, MN is a vessel of water in which tubes of various forms are immersed. The water will rise in the tubes A, B, C, to different altitudes m, n, o, inversely proportional to their diameters. If the tube B is broken at a, the water will not rise to the very top of it at a, but will stand at b, a little below the top, whatever be the length of the tube, or the diameter of its bore. If the tube be taken from the fluid and laid in a horizontal position, the water will recede from the end that was immersed. These two facts seem to countenance the opinion of Dr Jurin and other philosophers, that the water is attracted in the tube by the attraction of the annulus, or ring of glass, immediately above the cylinder of water. This hypothesis is sufficiently plausible; but supposing it to be true, the ring of glass immediately below the surface of the cylinder of fluid should produce an equal and opposite effect; and therefore the water instead of rising should be stationary, being influenced by two forces of an equal and opposite kind.

132. If a tube D composed of two cylindrical tubes of different bores be immersed in water with the widest part downwards, the water will rise to the altitude p, and if another tube E of the same size and form be plunged in the fluid with the smaller end downwards, the water will rise to the same height q as it did in the tube D. This experiment seems to be a complete refutation of the opinion of Dr Jurin, that the water is raised by the action of the annulus of glass above the fluid column; for since the annular surface is the same at q as at p, the same quantity of fluid ought to be supported in both tubes, whereas the tube E evidently raises much less water than D. But if we admit the supposition in art. 130, that the fluid is supported by the whole surface of glass in contact with the water, the phenomenon receives a complete explanation; for since the surface of glass in contact with the fluid in the tube E is much less than the surface in contact with it in the tube D, the quantity of fluid sustained in the former ought to be much less than the quantity supported in the latter.

133. When a vessel F, fig. 38, is plunged in water, and the lower part thereof filled by suction till the fluid enters the capillary tube G, whose bore is equal to the bore of the part F. In this experiment the portions of water on each side of the column F are supported by the pressure of the atmosphere on the surface of the water in the vessel MN; for if this vessel be placed in the exhausted receiver of an air-pump, these portions of water will not be sustained. Dr Jurin, indeed, maintains that these portions will retain their position in vacuo, but in his time the exhausting power of the air-pump was not sufficiently great to determine a point of so great nicety. The column tu, which is not sustained by atmospheric pressure, is kept in its position by the attraction of the water immediately around and above it, and the column Ft is supported by the attraction of the glass surface with which it is in contact. According to Dr Jurin's hypothesis, the column tu is supported by the ring of glass immediately above r, which is a very unlikely supposition.

134. The preceding experiment completely overturns the hypothesis of Dr Hamilton, afterwards revived by Dr Matthew Young. These philosophers maintained that the fluid was sustained in the tube by the lower ring of glass contiguous to the bottom of the tube, that this ring raises the portion of water immediately below it, and then other portions successively till the portion of water thus raised be in equilibrium with the attraction of the annulus in question. But if the elevation of the fluid were produced in this way, the quantity supported would be regulated by the form and magnitude of the orifice at the bottom of the tube; whereas it is evident from every experiment, that the cylinder of fluid sustained in capillary tubes has no reference whatever to the form of the lower annulus, but depends solely upon the diameter of the tube immediately above the elevated column of water.

135. If the experiments which we have now explained be performed in the exhausted receiver of an air-pump, the water will rise to the same height as when they are performed in an exhausted receiver.

---

1 Philosophical Transactions, No. 363, Art. 2. Capillary performed in air. We may therefore conclude, that the attraction of the water is not occasioned, as some have imagined, by the pressure of the atmosphere acting more freely upon the surface of the water in the vessel than upon the column of fluid in the capillary tube.

136. Numerous experiments have been made on the ascent of water and other fluids in capillary tubes. Among the earliest of these we may enumerate the experiments of M. Louis Carré, which were made previous to 1705. He found that in a tube twelve inches long water rose 10 lines, when the diameter of the tube was \( \frac{1}{2} \) of a line, 18 lines when its diameter was \( \frac{1}{4} \), and 30 lines when its diameter was \( \frac{1}{10} \) of a line.

The following are the results which he obtained with different fluids, and with a glass tube twelve inches in length:

| Names of Fluids | Diameter of the glass tube in parts of a line | Height of ascent in lines | |-----------------|-----------------------------------------------|--------------------------| | Water | | 10 | | Alcohol | | 3 | | Spirits of turpentine | \( \frac{1}{2} \) of a line | 4 | | Oil of tartar | | 5 | | Spirit of nitre | | 4 | | Oil of olives | | 5 |

Tube 9 Inches long.

| Water | \( \frac{1}{2} \) of a line | 10 | | Alcohol | | 4 |

Tube 15 Inches long.

| Water | \( \frac{1}{2} \) of a line | 20 | | Alcohol | | 12 |

Tube 5 Inches long.

| Water | \( \frac{1}{2} \) of a line | 27 | | Alcohol | | 12 nearly |

137. The following experiments were made by Mr. Benjamin Martin with a tube about \( \frac{1}{2} \) of an inch in diameter:

| Names of the Fluids | Height of Ascent in inches | Constant Number | |---------------------|----------------------------|-----------------| | Common spring water | 1.2 | .048 | | Spirit of urine | 1.1 | .044 | | Tincture of galls | 1.1 | .044 | | Recent urine | 1.1 | .044 | | Spirit of salt | 0.9 | .036 | | Ol. tart. per deliq.| 0.9 | .036 | | Vinegar | 0.95 | .038 | | Small beer | 0.9 | .036 | | Strong spirit of nitre | 0.85 | .034 | | Spirit of hartshorn | 0.85 | .034 | | Cream | 0.8 | .032 | | Skimmed milk | 0.8 | .032 | | Aquafortis | 0.75 | .030 | | Red wine | 0.75 | .030 | | White wine | 0.75 | .030 | | Ale | 0.75 | .030 | | Ol. sul. per campanam | 0.65 | .026 | | Oil of vitriol | 0.65 | .026 | | Sweet oil | 0.6 | .024 | | Oil of turpentine | 0.55 | .022 | | Geneva | 0.55 | .022 | | Rum | 0.5 | .020 | | Brandy | 0.5 | .020 | | White hard varnish | 0.5 | .020 | | Spirit of wine | 0.45 | .018 | | Tincture of mars | 0.45 | .018 |

138. To the preceding table as given by Mr. Martin we have added the constant number for each fluid, or the product of the altitude of the liquid, and the diameter of the tube (art. 130). By this number, therefore, we can find the altitude to which any of the preceding fluids will rise in a tube of a given bore, or the diameter of the bore when the altitude of the fluid is known; for since the constant number \( C = DA \) (art. 130.) we shall have \( D = \frac{C}{A} \) and \( A = \frac{C}{D} \).

Since the constant number, however, as deduced from the experiments of Martin, may not be perfectly correct, it would be improper to derive from it the diameter of the capillary bore when great accuracy is necessary. The following method, therefore, may be adopted as the most correct that can be given. Put into the capillary Method of tube a quantity of mercury, whose weight in troy grains is measuring \( W \); and let the length \( L \) of the tube which it occupies be accurately ascertained; then if the mercury be pure and at the temperature of 60° of Fahrenheit, the diameter of capillary tube,

\[ D = \sqrt{\frac{W}{L}} \times 0.019241, \]

the specific gravity of mercury being 13.580. The weight of a cubic inch of mercury being 3438 grains, and the solid contents of the mercurial column being \( D^2 L \times 0.7854 \), we shall have

\[ 1 : 3438 = D^2 L \times 0.7854 : W. \]

Hence (Geometry, Sect. IV. Theor. VIII.)

\[ D^2 L \times 0.7854 \times 3438 = W, \]

and dividing we have

\[ D^2 = \frac{W}{L \times 0.7854 \times 3438}, \]

or

\[ D = \sqrt{\frac{W}{L \times 0.7854 \times 3438}} = \frac{W}{L \times 0.7854 \times 3438}. \]

If the whole tube be filled with mercury, and if \( W \) be the difference in troy grams between its weight when empty, and when filled with mercury, the same theorem will serve for ascertaining the diameter of the tube. Should the temperature of the mercury happen to be 32° of Fahrenheit, its specific gravity will be 13.619, which will alter a very little the constant multiplier 0.019241.

139. Various experiments on the ascent of fluids in capillary tubes have been made by different individuals. The most important and recent of these were made by MM. Weltbrecht, Gellert, Lord Charles Cavendish, MM. Hally and Tremery, Sir David Brewster, and M. Gay Lussac.

The following are the results obtained by M. Weltbrecht in the ascent of water:

| Diameter of the tube in English inches | Height of ascent in inches | Constant quantity | |----------------------------------------|----------------------------|-------------------| | | 0.655 | 0.72 | 0.0432 | | | 0.045 | 0.95 | 0.04275 | | | 0.04 | 0.92 | 0.04140 | | | 0.05 | 0.85 | 0.0425 | | | 0.06 | 0.71 | 0.0426 | | | 0.08 | 0.53 | 0.0424 | | | 0.025 | 1.72 | 0.043 |

Mean, 0.04255

140. A series of experiments were made in 1740 by M. Of Gellert-Gellert, on the descent of melted lead in capillary tubes of glass. For this purpose he used the thinnest tubes, and heating them gradually before immersion in the lead, he obtained the following measures:

---

1 Mr. Martin found, that when capillary tubes, containing different fluids, were suspended in the sun for months together, the enclosed fluid was not in the least degree diminished by evaporation.

2 Comment. Acad. Petrop. 1736. M. Gellert likewise made experiments on the ascent of water in prismatic tubes, with a triangular and quadrangular bore, made of iron, and he found that they gave results analogous to those obtained with tubes of glass.

141. The most accurate experiments on the depression of mercury in capillary tubes are those made by Lord Charles Cavendish:

| Interior diameter of tube | Mercury in one inch of tube | Depression of the mercury | |---------------------------|-----------------------------|--------------------------| | 0.6 inches | 972 grains | 0.005 inches | | 0.5 | 675 | 0.007 | | 0.4 | 432 | 0.015 | | 0.35 | 331 | 0.025 | | 0.30 | 243 | 0.036 | | 0.25 | 169 | 0.050 | | 0.20 | 108 | 0.067 | | 0.15 | 61 | 0.092 | | 0.10 | 27 | 0.140 |

The constant quantity deduced by Dr Thomas Young from the preceding experiments is 0.015.

142. The following experiments were made by Dr Robinson, on the ascent of different fluids in a glass tube of a very slender bore, but its diameter is not mentioned.

| Solution of sal ammoniac | Height of ascent | |---------------------------|------------------| | Caustic volatile alkali | 8.07 inches | | Water | 6.25 | | Spirits of wine | 5.5 | | Oil of turpentine | 2.5 |

143. A series of careful experiments was made by MM. Lamy and Tremery's Hally and Tremery, at the request of La Place, on the ascent of water and oil of oranges in glass tubes; and also on the depression of mercury. The following results were obtained with water:

| Diameter of tube | Height of ascent | Constant quantity, or height for 1 millimetre | |------------------|------------------|-----------------------------------------------| | 2,000 millimetres| 6.75 millimetres | 13.50 | | 1.3333 | 10.00 | 13.333 | | 0.7500 | 18.50 | 13.875 |

Mean, 13.5693

With oil of oranges:

| Diameter of tube | Height of ascent | |------------------|------------------| | 2,000 | 3.400 | | 1.3333 | 5.000 | | 0.7500 | 9.000 |

With mercury:

| Diameter of tube | Depression | |------------------|------------| | 2,000 | 3.666 | | 1.3333 | 5.5 |

144. The very great discrepancy in the preceding results, obtained by very accurate and skilful observers, induced Sir David Brewster to repeat the experiments with an instrument constructed for the purpose, and to take such precautions, that he could always obtain the same results after repeated trials. The following description of the instrument which he used, and of the precautions which he adopted, will enable the reader to judge of the accuracy of the results.

Having obtained a glass tube 7.9 inches long, and of a uniform circular bore, he took a wire of a less diameter than the bore of the tube, and formed a small hook at one of its ends. This hook was fastened to the middle of a Capillary worsted thread, of such a size as, when doubled, to fill the Attraction bore of the tube. The wire was then passed through the tube, and the worsted thread drawn after it; and when the whole was plunged in an alkaline solution, the worsted thread was fixed at one end, and the tube was drawn backwards and forwards, till it was completely deprived, by its friction on the thread, of any grease or foreign matter which might have adhered to it. The tube and thread were then taken to clean water, and the same operation was repeated.

When the tube was thus perfectly cleaned, it was fixed vertically, by means of a level, in the axis of a piece of wood D (fig. 39.), supported by the arm AD, fixed upon a stand AB; and it was also furnished with an index mn, which was moveable to and from the extremity b. On the arm CE, moveable in a vertical direction by the nut C, was placed a glass vessel F, containing the fluid, and nearly filled with it. The nut C was then turned till the extremity b of the tube touched the surface of the fluid, which was indicated by the sudden rise of the liquor round its sides. The fluid then rose in the tube till it remained stationary, and the index mn was moved till its extremity n pointed out the exact position of the upper surface of the fluid. In this situation, the distance nb was a measure of the ascent of the liquid above its level in the vessel E. In order to ascertain, however, whether the fluid was stationary, in consequence of any obstruction in the tube, or of an equilibrium of the attracting forces, the vessel with the fluid was raised a little higher than its former position, by means of the nut C, and then depressed below it. If the fluid now rose a little above n, and afterwards sunk a little below it, so as always to rise and fall with facility and uniformity along with the surface of the fluid in the vessel, it was obvious that it suffered no obstruction in the tube, and that nb was the accurate measure of its height. By separating the extremity b of the tube from the surface of the fluid, the fluid always rises above n; but upon again bringing them into contact, the fluid resumes its position at n. If there should be any portion of fluid at the end b of the tube, when it is again brought in contact with the fluid surface, the water would rise around it before it had reached the general level, and therefore the height of the fluid obtained measuring from the end of the tube would be too small. In order to avoid this source of error, the index should have a projecting arm \( mn \), Fig. 40, carrying a screw &c., whose sharp point \( t \) can be easily brought on a level with the end \( b \) of the tube. When the extremity \( t \), therefore, which can always be kept dry, comes in contact with the fluid surface \( PQ \), the extremity \( b \) must also be exactly on the same level, even though the fluid had already risen around it. The tube was then cleaned, as formerly, for a subsequent observation. The results which were thus obtained for a great variety of fluids, and with a tube 0.0561 of an inch in diameter, are given in the following table:

| Names of Fluids | Height of Ascent, inches | Constant Quantity | |-----------------|-------------------------|------------------| | Water | 0.587 | 0.0327 | | Very hot water | 0.537 | 0.0301 | | Muriatic acid | 0.442 | 0.0248 | | Oil of boxwood | 0.427 | 0.0240 | | Oil of cassia | 0.420 | 0.0236 | | Nitrous acid | 0.413 | 0.0232 | | Oil of rapeseed | 0.404 | 0.0227 | | Castor oil | 0.403 | 0.0226 | | Nitric acid | 0.395 | 0.0222 | | Oil of spermaceti | 0.392 | 0.0220 | | Oil of almonds | 0.387 | 0.0217 | | Oil of olives | 0.387 | 0.0215 | | Balsam of Peru | 0.377 | 0.0212 | | Muriate of antimony | 0.373 | 0.0209 | | Oil of Rhodium | 0.366 | 0.0205 | | Oil of Pimento | 0.361 | 0.0203 | | Cajeput oil | 0.357 | 0.0200 | | Balsam of capivi| 0.357 | 0.0199 | | Oil of pennyroyal| 0.355 | 0.0199 | | Oil of thyme | 0.354 | 0.0199 | | Oil of bricks distilled from spermaceti oil | 0.354 | 0.0199 | | Oil of caraway seeds | 0.353 | 0.0198 | | Oil of rhue | 0.353 | 0.0198 | | Oil of spearmint| 0.351 | 0.0197 | | Balsam of sulphur | 0.349 | 0.0196 | | Oil of sweet fennel seeds | 0.349 | 0.0195 | | Oil of hyssop | 0.349 | 0.0195 | | Oil of rosemary | 0.344 | 0.0193 | | Oil of bergamot | 0.343 | 0.0192 | | Oil of amber | 0.343 | 0.0192 | | Oil of anise seeds | 0.342 | 0.0192 | | Oil of Barbadoes tar | 0.341 | 0.0191 | | Laudanum | 0.340 | 0.0191 | | Oil of cloves | 0.334 | 0.0187 | | Oil of turpentine | 0.333 | 0.0187 | | Oil of lemon | 0.333 | 0.0187 | | Oil of lavender | 0.328 | 0.0184 | | Oil of camomile | 0.327 | 0.0184 | | Oil of peppermint | 0.327 | 0.0184 | | Oil of sassafras | 0.327 | 0.0184 | | Highland whisky | 0.327 | 0.0184 | | Brandy | 0.326 | 0.0183 | | Oil of wormwood | 0.326 | 0.0183 | | Oil of dill seed | 0.324 | 0.0182 | | Oil of ambergrase | 0.323 | 0.0181 | | Genuine oil of juniper | 0.321 | 0.0180 | | Oil of nutmeg | 0.320 | 0.0180 | | Alcohol | 0.317 | 0.0178 | | Oil of savine | 0.310 | 0.0174 | | Ether | 0.285 | 0.0160 | | Oil of wine | 0.273 | 0.0153 | | Sulphuric acid | 0.200 | 0.0112 |

The constant quantity in English inches, as deduced from these two experiments, is 0.04622.

Experiments with Alcohol.

| Diameter of the tube | Height of ascent above lowest point of concavity | Density of alcohol | |----------------------|--------------------------------------------------|-------------------| | 1.29441 millim. | 9.18235 millim. | 0.81961 | | 1.90381 | 6.08397 | 0.81961 | | 1.29441 | 9.30079 | 0.8595 | | 1.29441 | 9.99727 | 0.94153 | | 10.508 | 0.3835 | 0.81347 |

The temperature of the alcohol was 8°.5 centigrade, and the constant quantity for the two first experiments, reduced to English inches, is 0.01815, which agrees remarkably with 0.0178, the constant quantity in Sir David Brewster's experiments.

Experiments with Oil of Turpentine.

| Diameter of tube | Height of fluid | Density | |------------------|----------------|--------| | 1.29441 millim. | 9.95159 | 0.869458 |

This result also coincides very nearly with that of Sir David Brewster.

146. The following table contains a general view of the General results obtained by different philosophers, from the ascent results of water in capillary tubes.

| Names of observers | Constant quantity, in English inches | |--------------------|-------------------------------------| | Sir Isaac Newton | 0.020 | | MM. Haity and Tremery | 0.021 | | M. Carré, mean of three observations | 0.022 | | M. Hallstrom | 0.026 | | Sir David Brewster | 0.033 | | Muschenbroek | 0.039 | | M. Weibrecht, average of his results | 0.042 | | M. Gay-Lussac, average of 2 observations | 0.046 | | Benjamin Martin | 0.048 | | Mr Atwood | 0.053 | | James Bernoulli | 0.064 |

Throwing aside the measure of James Bernouilli as obviously erroneous, we obtain 0.035 as the general average the discrepancy between the preceding means; but the difference between this and the extreme measures of Newton and Atwood is so great, that there must be some cause, different from an error of observation, to which it is owing. The difference between the results obtained by Sir David Brewster and M. Gay-Lussac, made with nice instruments founded on the same principle, leads to the same conclusion. La Place indeed has ascribed, and we think justly, these differences to the greater or less degree of humidity on the sides of the tubes; and he informs us that Guy Lussac made his experiments with tubes very much wetted. Here, then, we have at once the cause of the difference above mentioned, because the experiments of Sir David Brewster... were made with a tube carefully cleaned and dried after each experiment. A dry tube must necessarily raise the water to a less height than a wet one, and the difference must increase as the diameter of the tube employed is diminished. If we conceive a tube, indeed, with an exceedingly small bore, wetted over the whole of its interior, in the slightest degree, the two inner surfaces of the film would nearly meet in the axis, and the height of ascent would be infinite, or as high as the tube was long.

147. From these observations, the reader will be already prepared to draw the conclusion, that the ascent of fluids in glass tubes is a very equivocal measure of the force of capillary attraction, independent of its being applicable only to the single substance of glass.

With the view of removing this objection, Sir David Brewster long ago constructed an instrument, the object of which was to measure, upon an optical principle, the diameter of the circle of fluid which any cylindrical solid raises by capillary attraction above its general level. Thus let MN (fig. 41.) be the plan of a vessel filled with fluid, and A the section of a vertical cylinder of well cleaned and well dried glass, or any other substance not porous. This solid A will raise the fluid to a certain height around it, elevating a circular portion CD of the fluid above the general level; and it is manifest that the diameter CD of this elevated portion will be proportioned to the height of the fluid round the sides of the cylinder, or to the capillary force by which it is raised. In order to measure the diameter of this circle of fluid, a micrometer carries a small vertical frame along the edge FG of the vessel. Along this frame are stretched two fine parallel wires, whose images can be seen by reflection from the surface of the fluid, by an eye on the side PQ of the vessel, aided by a microscope with a distant focus. When the image of these wires is seen by reflection from any part of the fluid surface without the circle CD, it will suffer no change of form; but when it is seen by reflection from any portion of the elevated portion CD, the fibres will appear disturbed, and will indicate, by their return to the rectilineal form of accurate parallelism, the apparent termination of the circle CD. The same observation is made on the other side of A, at the boundary D, and a measure is thus obtained of the diameter of the circle CD, by means of the micrometer screw, by which the microscope on the side PQ, and the wire frame on the side FG are moved (being fixed to the same frame) along the sides of the vessel. In this way solids of all kinds may be used, and their exterior or acting surfaces may be easily cleared from grease and other adhering substances.

This apparatus may be improved by using two cylinders A, B in place of one, and by moving one of them, suppose B, from the other A till the two elevated circular portions CD, DE disturb the images of the wires, seen by reflection from the intermediate point at D.

148. When water is made to pass through a capillary tube of such a bore that the fluid is discharged only by successive drops; the tube, when electrified, will furnish a constant and accelerated stream, and the acceleration is proportional to the smallness of the bore. A similar effect may be produced by employing warm water. Sir John Leslie found that a jet of warm water rose to a much greater height than a jet of cold water, though the water in both cases moved through the same aperture, and was influenced by the same pressure. A syphon also, which discharged cold water only by drops, furnished warm water in an uninterrupted stream.

149. Such are the leading phenomena of capillary tubes. Capillary attraction, considered; and while it furnishes us with a very beautiful experiment, it confirms the reasoning by which we have accounted for the elevation of fluids in cylindrical canals. Let ABEF and CDEF be two places of plate glass with smooth and clean surfaces, having their sides EF joined together with wax, and their sides AB, CD kept a little distance by another piece of wax W, so that their interior surfaces, whose common intersection is the line EF, may form a small angle. When this apparatus is immersed in a vessel MN full of water, the fluid will rise in such a manner between the glass planes as to form the curve Dgome, which represents the surface of the elevated water. By measuring the ordinates mn, op, &c. of this curve, and also its abscisse Fn, Fp, &c. Mr Hawksbee found it to be the common Apollonian hyperbola, having for its asymptotes the surface DE of the fluid, and EF the common intersection of the two planes. The following are the results which he obtained when the inclination of the planes was 20°:

| Distances from the touching ends of the planes | Heights of the water at the preceding distances | |-----------------------------------------------|------------------------------------------------| | 13 inches | 1 | | 9 | 2 | | 7 | 3 | | 6 | 3½ | | 5 | 5 | | 4 | 6½ | | 3 | 9 | | 2½ | 12 | | 2 | 15½ | | 1¾ | 18 | | 1¼ | 21½ | | 1¼ | 27½ | | 1 | 35 | | ¾ | 50 | | ½ | 76 |

By repeating these observations at inclinations of 40°, and at various other angles, Mr Hawksbee found that the curve was an exact hyperbola in all directions of the planes. To the very same conclusion we are led by the principles already laid down; for as the distance between the plates diminishes at every point of the curve Dgome from D towards F, the water ought to rise higher at o than at q, still higher at m, and highest of all at F, where the distance between the plates is a minimum. To illustrate this more clearly, let ABEF and CDEF (fig. 43.) be the same plates of glass (inclined at a greater angle for the sake of distinctness), and let FmgD, and FosB be the curves which bound the surface of the elevated fluid. Then, since the altitudes of the water in capillary tubes are in- versely as their diameters, or the distances of their opposite sides, the altitudes of the water between two glass plates, should at any given point be inversely as the distances of the plates at that point. Now, the distance of the plates at the point \( m \) is obviously \( mo \), or its equal \( np \), and the distance at \( q \) is \( qs \) or \( rt \); and since \( mn \) is the altitude of the water at \( m \), and \( qr \) its altitude at \( q \), we have

\[ \frac{em}{mn} = \frac{qr}{rt} \]

but (Geometry, Sect. IV. Theor. XVII.) \( En : Er = np : rt \); therefore \( mn : qr = Fn : Fr \); that is, the altitudes of the fluid at the points \( m, q \), which are equal to the abscissae \( Fn, Fr \) (fig. 42.) are proportional to the ordinates \( qr, mn \), equal to the abscissae \( Fn, Fr \), in fig. 42. But in the Apollonian hyperbola the ordinates are inversely proportional to their respective abscissae; therefore the curve \( Dqmf \) is the common hyperbola. As the plates are infinitely near each other at the apex \( F \), the water will evidently rise to that point, whatever be the height of the plates.

150. Mr Hawksbee extended his experiments to plates of glass placed parallel to each other, and separated to different distances, and he obtained the following results:

| Distance of plates | Height of Ascent | Constant quantity | |--------------------|-----------------|------------------| | 0.0625 of an inch | 0.166 of an inch | 0.0104 | | 0.03125 | 0.333 | 0.0104 | | 0.015625 | 0.666 | 0.0104 | | 0.007802 | 1.333 | 0.0104 |

The following experiments on the same subject have been more recently made by M. Monge, MM. Haüy and Tremery, and M. Gay-Lussac.

In those made by M. Monge, the plates were first cleaned with caustic alkali, and well washed. Their degree of separation was ascertained by silver wires of different thicknesses, and the fluid used was the filtered water of the Seine. The following were the results:

| Distance of plates in parts of a line | Height of ascent | Constant quantity | |---------------------------------------|-----------------|------------------| | \( \frac{1}{3} \) or 0.0101 inch | 15.5 lines | 0.1565 | | \( \frac{1}{5} \) | 33.5 | 0.2278 | | \( \frac{1}{8} \) | 74 | 0.222 |

The following result was obtained by MM. Haüy and Tremery:

| Distance of plate | Height of ascent | Const. quant. | |-------------------|-----------------|---------------| | 1 millimetre | 6.5 millimetre | 6.5 |

The following measures were obtained by M. Gay-Lussac, with plates of glass ground perfectly flat:

| Distance of plates above lowest point of concavity | Temp.-coeff. | |----------------------------------------------------|-------------| | 1.069 millimetre | 13.574 | 16° |

Here the constant quantity is 14.51, or 0.02251, when reduced to English inches for a distance of \( \frac{1}{16} \)th of an inch.

151. The phenomena which we have been considering are all referrible to one simple fact, that the particles of glass have a stronger attraction for the particles of water than the particles of water have for each other. This is the case with almost all other fluids except mercury, the particles of which have a stronger attraction for each other than for glass. When capillary tubes, therefore, are plunged in this fluid, a new series of phenomena present themselves to our consideration. Let MN (fig. 44) be a vessel full of mercury. Plunge into the fluid Capillary the capillary tube CD, and the Attraction, mercury, instead of rising in the tube, will remain stationary at E, its depression below the level surface AB being inversely proportional to the diameter of the bore. This was formerly ascribed to a repulsive force supposed to exist between mercury and glass, but we shall presently see that it is owing to a very different cause.

152. That the particles of mercury have a very strong Mercury attraction for each other, appears from the globular form has a which a small portion of that fluid assumes; and from the stronger resistance which it opposes to any separation of its parts, attraction If a quantity of mercury is separated into a number of minute parts, all these parts will be spherical; and if two of these spheres be brought into contact, they will instantly glass rush together, and form a single drop of the same form. There is also a very small degree of attraction existing between glass and mercury; for a globule of the latter very readily adheres to the lower surface of a plate of glass. Now, suppose a drop of water laid upon a surface anointed with Cause of grease, to prevent the attraction of cohesion from reducing the depression to a film of fluid, this drop, if very small, will be spherical. If its size is considerable, the gravity of its parts will make it spheroidal, and as the drop increases in magnitude, it will become more and more flattened at its poles, like AB in fig. 45.

The drop, however, will still retain its convexity at the circumference, however oblate be the spheroid into which it is moulded by the force of gravity. Let two pieces of glass \( oAm, pBn \) be now brought in contact with the circumference of the drop; the mutual attraction between the particles of water which enabled it to preserve the convexity of its circumference will yield to their superior attraction for glass; the space \( m, n, o, p \), will be immediately filled; and the water will rise on the sides of the glass, and the drop will have the appearance of AB in fig. 46. If the drop AB (fig. 45.) be now supposed mercury instead of water, it will also, by the gravity of its parts, assume the form of an oblate spheroid; but when the pieces of glass \( oAm, pBn \) are brought close to its periphery, their attractive force upon the mercurial particles is not sufficient to counteract the mutual attraction of these particles; the mercury, therefore, retains its convexity at the circumference, and assumes the form exhibited in fig. 47. The small spaces \( op \) being filled by the pressure of the superincumbent fluid, while the spaces below \( m, n \), still remain between the glass and the mercury. Now if the two plates of glass A, B be made to approach each other, the depressions \( m, n \) will still continue, and when the distance of the plates is so small that these depressions or indentations meet, the mercury will sink between the plates, and its descent will continue as the pieces of glass approach. Hence the depression of the mercury in capillary tubes becomes very intelligible. If two glass planes forming a small angle, as in fig. 42, be immersed in a vessel of mercury, the fluid will sink below the surface of the mercury in the vessel, and form an Apollonian hyperbola like \( DqF \), having for its asymptotes the common intersection of the planes and the surface of mercury in the vessel.

153. The depression of mercury in capillary tubes is evidently owing to the greater attraction that subsists be- The capillary between the particles of mercury and those of glass. The attraction difference between these two attractions, however, arises from an imperfect contact between the mercury and the capillary tube, occasioned by the interposition of a thin coating of water which generally lines the interior surface of the tube, and weakens the mutual action of the glass and mercury; for this action always increases as the thickness of the interposed film is diminished by boiling. In the experiments which were made by Laplace and Lavoisier on barometers, by boiling the mercury in them for a long time, the convexity of the interior surface of the mercury was often made to disappear. They even succeeded in rendering it concave, but could always restore the convexity by introducing a drop of water into the tube. When the ebullition of the mercury is sufficiently strong to expel all foreign particles, it often rises to the level of the surrounding fluid, and the depression is even converted into an elevation.

154. Between mercury and water there is likely to be some fluid in which the attraction of the glass for its particles is nearly equal to half the attraction of the fluid for itself. Sir David Brewster has observed that iodine dissolved in chloride of sulphur approximates to this condition; but not having the chloride by itself, he could not observe whether or not the effect is produced or influenced by the iodine. If it is, then a solution may be obtained, in which the above condition is perfect. The solution of the iodine already mentioned scarcely rises on the sides of the glass ball which contains it. (See 162.)

155. As most philosophers seem to agree in thinking that all the capillary phenomena are referable to the cohesive attraction of the superficial particles only of the fluid, a variety of experiments has been made in order to determine the force required to raise a horizontal solid surface from the surface of a fluid. Mr Achard found that a disc of glass, 14 French inches in diameter, required a weight of 91 French grains to raise it from the surface of the water at 69° Fahrenheit, which is only 37 English grains for each square inch. At 441 of Fahrenheit the force was 1/4 greater, or 391 grains, the difference being 1/4 for each degree of Fahrenheit. From these experiments Dr Young concludes that the height of ascent in a tube of a given bore, which varies in the duplicate ratio of the height of adhesion, is diminished about 1/4 for every degree of Fahrenheit that the temperature is raised above 50°; and he conjectures that there must have been some considerable source of error in Achard's experiments, as he never found this diminution to exceed 1/8. According to the experiments of Dutton, the force necessary to elevate the solid, or the quantity of water raised, is equal to 44.1 grains for every square inch.

156. According to the experiments of Morveau, the force necessary to elevate a circular inch of gold from the surface of mercury is 446 grams; a circular inch of silver, 429 grams; a circular inch of tin, 418 grams; a circular inch of lead, 397 grams; a circular inch of bismuth, 372 grams; a circular inch of zinc, 204 grams; a circular inch of copper, 142 grams; a circular inch of metallic antimony, 126; a circular inch of iron, 115 grams; and a similar surface of cobalt required 8 grams. The order in which these metals are arranged is the very order in which they are most easily amalgamated with mercury.

The most recent experiments on the adhesion of surfaces to fluids have been made by M. Gay-Lussac, who obtained the following results with a circular plate of glass 118.356 millimetres in diameter:

| Names of fluids | Weight necessary to raise the plate from the glass | Specific gravity | |-----------------|--------------------------------------------------|-----------------| | Water | 59.40 grammes | 1.000 | | Alcohol | 31.08 | 0.8196 | | Alcohol | 32.87 | 0.8595 | | Oil of turpentine | 37.152 | 0.9415 |

With a copper disc, 116.604 millimetres in diameter, the capillary weight necessary to raise it from water, at the temperature of 18°.5 centigrade, was 57.945 grammes, differing very little, if at all, from glass, for the diminution of weight may be explained by the circumstance of the copper disc being nearly two millimeters less in diameter than the glass. In these experiments the discs were suspended from the scale of a balance, and the weights in the other scale successively increased till the force of adhesion was overcome at the instant when the disc detached itself from the fluid surface.

157. A number of experiments on the adhesion of fluids have been lately made by Count Rumford, which authorize him to conclude, that on account of the mutual adhesion of the particles of fluid, a pellicle or film is formed at Rumford on the superior and inferior surfaces of water, and that the force of the film to resist the descent of bodies specifically heavier than the fluid increases with the viscosity of the water. He poured a stratum of sulphuric ether upon a quantity of water, and introduced a variety of bodies specifically heavier than water into this compound fluid. A sewing needle, granulated tin, and small globules of mercury, descended through the ether, but floated upon the surface of the water. When the eye was placed below the level of the aqueous surface, the floating body, which was a spherule of mercury, seemed suspended in a kind of bag a little below the surface. When a larger spherule of mercury was employed, about the 40th or 50th of an inch in diameter, it broke the pellicle and descended to the bottom. The same results were obtained by using essential oil of turpentine or oil of olives instead of ether. When a stratum of alcohol was incumbent upon the water, a quantity of very fine powder of tin thrown upon its surface, descended to the very bottom, without seeming to have met with any resistance from the film at the surface of the water. This unexpected result Count Rumford endeavours to explain by supposing that the aqueous film was destroyed by the chemical action of the alcohol. In order to ascertain with greater accuracy the existence of a pellicle at the surface of the water, Count Rumford employed a cylindrical glass vessel 10 inches high and 1½ inch in diameter, and filled it with water and ether as before. A number of small bodies thrown into the vessel descended through the ether and floated on the surface of the water. When the whole was perfectly tranquil, he turned the cylinder three or four times round with considerable rapidity in a vertical position. The floating bodies turned round along with the glass, and stopped when it was stopped; but the liquid water below the surface did not at first begin to turn along with the glass; and its motion of rotation did not cease with the motion of the vessel. From this Count Rumford concludes that there was a real pellicle at the surface of the water, and that this pellicle was strongly attached to the sides of the glass, so as to move along with it. When this pellicle was touched by the point of a needle, all the small bodies upon its surface trembled at the same time. The apparatus was allowed to stand till the ether had entirely evaporated, and when the pellicle was examined with a magnifier, it was in the same state as formerly; and the floating bodies had the same relative positions.

In order to show that a pellicle was formed at the inferior surface of water, Count Rumford poured water upon mercury, and upon that a stratum of ether. He threw into the vessel a spherule of mercury about one-third of a line in diameter, which being too heavy to be supported by the pellicle at the superior surface of the water, broke it, and descending through that fluid, was stopped at its inferior surface. When this spherule was moved, and even compressed with a feather, it still preserved its spherical form, and refused to mix with the mass of mercury. When the viscosity of the water was increased by the infusion of gum-arabic, much larger spherules were supported by the Capillary pellicle. From the very rapid evaporation of ether, and its inability to support the lightest particles of a solid upon its surface, Count Rumford very justly concludes, that the mutual adhesion of its particles is very small.

158. The approach of two floating bodies has been ascribed by some to their mutual attraction, and by others to the attraction of the portions of fluid that are raised round each by the attraction of cohesion. Dr Young, however, observes, that the approach of the two floating bodies is produced by the excess of the atmospheric pressure on the remote sides of the solids, above its pressure on their neighbouring sides; or, if the experiments are performed in a vacuum, by the equivalent hydrostatic pressure or suction derived from the weight and immediate cohesion of the intervening fluid. This force varies alternately in the inverse ratio of the square of the distance; for when the two bodies approach each other, the altitude of the fluid between them is increased in the simple inverse ratio of the distance; and the mean action, or the negative pressure of the fluid on each particle of the surface, is also increased in the same ratio. When the floating bodies are surrounded by a depression, the same law prevails, and its demonstration is still more simple and obvious.

159. A different view of the subject has been given by Monge, who made a number of accurate experiments on the subject, and deduced from them the following laws:

1. If two floating bodies, capable of being wetted with the fluid on which they float, are placed near each other, they will approach as if mutually attracted.

In order to explain this law, let AB, CD (fig. 48.) be two suspended plates of glass, placed at such distance that the point H where the two portions of elevated fluid meet, is on a level with the rest of the water, the two plates will remain stationary and in perfect equilibrium. But if they are brought nearer one another, as in fig. 49, the water will rise between them to a point H above the level, and by a nearer approximation, to the point G. The water thus elevated, acting like a curved chain hung to the two plates, attracts the sides of the plates, and brings them together in a horizontal direction. The very same thing takes place with the floating bodies A, B placed at such a distance that the water rises between them above its level, and hence these bodies will approach by the attraction of the fluid on their inner sides.

2. When the two floating bodies A, B, are not capable of being wetted, they will approach each other as if mutually attracted, when they are placed near one another.

In this case the fluid is depressed between them below its natural level H, and the two bodies are pressed inwards or towards each other, which pressure being greater than the pressure outwards of the fluid between them, they will approach each other by the action of the difference of these pressures.

3. If one of the bodies A, is capable of being wetted, and the other B not, as shown in the figure, they will recede from each other as if mutually repelled.

As the fluid rises round A, and is depressed round B, the depression round B will not be equal all round, and hence the body B, being placed as if on an inclined plane, it will move to the right hand when the pressure is the least.

In this last case, La Place was led by theory to believe that when B is placed very near A, the repulsion will be converted into attraction. M. Haiy tried this experiment with planes of ivory and talc, the former being incapable of being wetted with water and the latter not; and he found, in conformity with La Place's prediction, that at a certain short distance the talc moved suddenly into contact with the ivory.

160. The phenomena of attraction and repulsion exhibited between small lighted wicks, swimming in a basin of oil, and the motions of floating evaporable substances like camphor, and also of potassium and light substances, such as cork, impregnated with ether, have been sometimes treated under this head. The first of these classes of phenomena arise from an unbalanced pressure upon the floating wick, arising from a difference of temperature of different parts of the oil, and the movements of the second class arise from the reaction of the currents of vapour which flow from the floating substances. A full account of these phenomena will be found in the Edinburgh Transactions (vol. iv. p. 44), in the Mémoires Presentées à L'Institut (tom. i. p. 129), and the more recent observations of Matteucci, in the Annales de Chimie (June 1833, tom. iii. p. 216-219.)

Theory of Capillary Attraction.

161. Clairaut was the first mathematician who attempted to analyze the forces which contribute to the ascent of fluids in capillary tubes. After pointing out the insufficiency of preceding theories, he gives an analysis of the different forces which contribute to the suspension of fluids in capillary tubes.

Let ABCDEFGH (fig. 53.) be the section of a capillary tube, MNP the surface of the water in the vessel, T the height of its ascent, viz. the concave surface of the fluid column, and IKLM an indefinitely small column of fluid reaching to the surface at M. Now the column ML is solicited by the force of gravity which acts through the whole extent of the column, and by the reciprocal attraction of the molecule, which, though they act the same in all the points of the column, only exhibit their effects towards the extremity M. If any particle e is taken at a less distance from the surface than the distance at which the attraction of the liquid generally terminates, and if mn is a plane parallel to MN, and at the same distance from the particle e,

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1 Théorie de la Figure de la Terre, chap. x. Paris, 1743, 1803. between the planes MN, m n. The water, however, below m n, will attract the particle downwards, and this effect will take place as far as the distance where the attraction ceases.

The column IK, on the other hand, which is in a state of equilibrium with ML, is acted upon by the force of gravity through the whole extent of the column, also by other forces at the upper and lower extremities of the tube. The forces exerted at the upper part of the column are, the attraction of the tube upon the particles of water, and the reciprocal attraction of these particles; but as every particle is as much drawn upwards as downwards by the first of these forces, the consideration of it may be dropped.

In order to estimate the other force, let a horizontal plane VX touch the concavity at I, a particle p, situated infinitely near to I, is attracted by all the particles above VX, and by all below it whose sphere of activity comprehends that particle; and as the particles above p are fewer than those below it, the result of these forces must be a force acting downwards.

In order to estimate the value of the forces which act at the lower end O of the tube, let us suppose that the tube has a prolongation to the bottom of the vessel, formed of matter of the same density as the water. Let a particle R be situated a little above the extremity of the tube, and another Q as much below that extremity, they will be equally acted upon by the water above that place, and by the water between the fictitious prolongation of the tube, and therefore these forces will destroy one another.

By applying to the case of the particle R the same reasoning that was used for the particle e, it will appear that the result of its attraction by the tube is an attraction upwards. The particle R is likewise attracted downwards by the supposed prolongation of the tube, and the difference between these is the real effect. The other particle Q is also drawn upwards by the tube with the same force as R, since, by the hypothesis, it is as far distant from the points D, G, as the particle R is from the points d, g, where, with respect to it, the real attraction of the tube commences. The particle Q is attracted also downwards, by the supposed prolongation of the tube, and the difference of these actions is the real effect. Hence the double of this force is the sum of all the forces that act at the lower part of the tube. These forces, when combined with those exerted at the top of the tube, and with the force of gravity, give the total expression, which should be combined with that of the forces with which the column ML is actuated.

The formula obtained by Clairaut for the altitude L i, fig. 53, is

\[ \int_{L}^{i} (2Q - Q') dx [b, x] + \int_{L}^{i} dx [b, x, Q, Q'] \]

in which Q is the intensity of the attraction of the glass, Q' the intensity of the attraction of the water, b the interior radius of the tube, and p the force of gravity.

Clairaut then observes, that there is an infinite number of possible laws of attraction which will give a sensible quantity for the elevation of the fluid L i above the level MN, when the diameter of the tube is very small, and a quantity next to nothing when the diameter is considerable; and he remarks, that we may select the law which gives the inverse ratio between the diameter of the tube and the height of the liquid, conformable to experiment.

162. It follows from the preceding formula, that if any solid AB possesses half the attracting power of the fluid CD, the surface of the fluid will remain horizontal; for the attraction being represented by DA, DE, and DC, DA and DE may be combined into DB, and DB and DC into DE, which is vertical. The water will therefore not be raised, since the surface of a fluid at rest must be perpendicular to the resulting direction of all the forces which act upon it.

When the attracting power of the solid is more than half as great, the resultant of the forces will be GF in fig. 55, and therefore the fluid must rise towards the solid, in order to be perpendicular to GF. When the attractive power of the solid is less than that of the fluid, the resultant will be HF in fig. 55; and therefore, as in the case of mercury, the surface must be depressed, in order to be perpendicular to the force.

163. The subject of capillary attraction has more recently occupied the attention of the late Marquis de la Place, who published his first remarks on the subject in 1806. In 1807 he published a supplement to his theory, in which he compares his formula with the experiments of Gay-Lussac and others.

In the first treatise published by M. La Place, his method of considering the phenomena was founded on the form of La Place's Theory. The conditions of equilibrium of this fluid in an infinitely narrow canal, resting by one of its extremities upon this surface, and by the other on the horizontal surfaces of an indefinite fluid, in which the capillary tube was immersed. In his supplement to that treatise, he has examined the subject in a much more popular point of view, by considering directly the forces which elevate and depress the fluid in this space. By this means, he is conducted easily to several general results, which it would have been difficult to obtain directly by his former method. Of this method we shall endeavour to give as clear a view as possible.

164. Let AB, fig. 56, be a vertical tube whose sides are perpendicular to its base, and which is immersed in a fluid of the capillary forces. Capillary formed of a film of ice, so as to have no action on the fluid attraction which it contains, and not to prevent the reciprocal action &c., which takes place between the particles of the first tube AB and the particles of the fluid. Now, since the fluid in the tubes AE, CD, is in equilibrium, it is obvious that the excess of pressure of the fluid in AE is destroyed by the vertical attraction of the tube and of the fluid upon the fluid contained in AB. In analyzing these different attractions, La Place considered first those which take place under the tube AB. The fluid column BE is attracted, 1. by itself; 2. by the fluid surrounding the tube BE. But these two attractions are destroyed by the similar attractions experienced by the fluid contained in the branch DC, so that they may be entirely neglected. The fluid in BE is also attracted vertically by the fluid in AB; but this attraction is destroyed by the attraction which it exercises in the opposite direction upon the fluid in BE, so that these balanced attractions may likewise be neglected. The fluid in BE is likewise attracted vertically upwards by the tube AB, with a force which we shall call Q, and which contributes to destroy the excess of pressure exerted upon it by the column BF raised in the tube above its natural level.

Now, the fluid in the lower part of the round tube AB is attracted, 1. By itself; but as the reciprocal attractions of a body do not communicate to it any motion if it is solid, we may, without disturbing the equilibrium, conceive the fluid in AB frozen. 2. The fluid in the lower part of AB is attracted by the interior fluid of the tube BE, but as the latter is attracted upwards by the same force, these two actions may be neglected as balancing each other. 3. The fluid in the lower part of BE is attracted by the fluid which surrounds the ideal tube BE, and the result of this attraction is a vertical force acting downwards, which we may call — Q', the contrary sign being applied, as the force is here opposite to the other force Q. As it is highly probable that the attractive forces exercised by the glass and the water vary according to the same function of the distance, so as to differ only in their intensities, we may employ the constant co-efficients ε, ε' as measures of their intensity, so that the forces Q and — Q' will be proportional to ε, ε'; for the interior surface of the fluid which surrounds the tube BE, is the same as the interior surface of the tube AB. Consequently, the two masses, viz. the glass in AB, and the fluid round BE differ only in their thickness; but as the attraction of both these masses is insensible at sensible distances, the difference of their thicknesses, provided their thicknesses be sensible, will produce no difference in the attractions. 4. The fluid in the tube AB is also acted upon by another force, namely, by the sides of the tube AB in which it is inclosed. If we conceive the column FB divided into an infinite number of elementary vertical columns, and if at the upper extremity of one of these columns we draw a horizontal plane, the portion of the tube comprehended between the plane and the level surface BC of the fluid will not produce any vertical force upon the column; consequently, the only active vertical force is that which is produced by the ring of the tube immediately above the horizontal plane. Now, the vertical attraction of this part of the tube upon BE, will be equal to that of the entire tube upon the column BE, which is equal in diameter, and similarly placed. This new force will therefore be represented by + Q. In combining these different forces, it is manifest that the fluid column BF is attracted upwards by the two forces + Q, + Q, and downwards by the force — Q'; consequently the force with which it is raised upwards will be 2Q — Q'. If we represent by V the volume of the column BE, by D its intensity, and by g the force of gravity, then gDV will represent the weight of the elevated column; but as this weight is in equilibrium with the forces by which it is elevated, we have the following equation:

\[ gDV = 2Q - Q'. \]

If the force 2Q is less than Q', then V will be negative, and the fluid will sink in the tube; but as long as 2Q is greater than Q', V will be positive, and the fluid will rise above its natural level; as was long before shewn by M. Clairaut.

Since the attractive forces, both of the glass and the fluid, are insensible at sensible distances, the surface of the tube AB will act sensibly only on the column of fluid immediately in contact with it. We may therefore neglect the consideration of the curvature, and consider the inner surface as developed upon a plane. The force Q will therefore be proportional to the width of this plane, or what is the same thing, to the interior circumference of the tube. Calling c, therefore, the circumference of the tube, we shall have \( Q = \varepsilon c \); ε being a constant quantity, representing the intensity of the attraction of the tube AB upon the fluid, in the case where the attractions of different bodies are expressed by the same function of the distance. In every case, however, ε expresses a quantity dependent on the attraction of the matter of the tube, and independent of its figure and magnitude. In like manner we shall have \( Q' = \varepsilon' c \); ε' expressing the same thing with regard to the attraction of the fluid for itself, that ε expressed with regard to the attraction of the tube for the fluid. By substituting these values of Q, Q', in the preceding equation, we have

\[ gDV = c(2\varepsilon - \varepsilon'). \]

If we now substitute, in this general formula, the value of ε in terms of the radius if it is a capillary tube, or in terms of the sides if the section is a rectangle, and the value of V in terms of the radius and altitude of the fluid column, we shall obtain an equation by which the heights of ascent may be calculated for tubes of all diameters, after the height, belonging to any given diameter, has been ascertained by direct experiment.

165. In the case of a cylindrical tube, let ε represent the Applicatio of the circumference to the diameter, h the height of the fluid column reckoned from the lower point of the meniscus, q the mean height to which the fluid rises, or the height at which the fluid would stand if the meniscus were to fall down and assume a level surface, then we have \( \pi r^2 \) for the solid contents of a cylinder of the same height and radius as the meniscus, and as the meniscus, added to the solid contents of the hemispheres of the same radius, must be equal to \( \pi r^2 \), we have \( \pi r^2 = \frac{2\pi r^3}{3} \), or \( \frac{\pi r^3}{3} \), for the solid contents of the meniscus. But since \( \frac{\pi r^3}{3} = \pi r^3 \times \frac{r}{3} \), it follows that the meniscus \( \frac{\pi r^3}{3} \) is equal to a cylinder whose base is \( \pi r^2 \), and altitude \( \frac{r}{3} \). Hence, we have

\[ q = h + \frac{r}{3}; \]

or what is the same thing, the mean altitude q in a cylinder is always equal to the altitude h of the lower point of the concavity of the meniscus increased by one-third of the radius, or one-sixth of the diameter of the capillary tube. Now, since the contour c of the tube \( = 2\pi r \), and since the volume V of water raised is equal to \( q \times \pi r^2 \), we have, by substituting these values in the general formula,

\[ gDq\pi r^2 = 2\pi r(2\varepsilon - \varepsilon'), \quad (\text{No. 1}). \]

spillary and dividing by \( \pi r \) and \( g D \), we have,

\[ r q = 2 \frac{2 \epsilon - \epsilon'}{g D} \text{ and } q = 2 \frac{2 \epsilon - \epsilon'}{g D} \times \frac{1}{r}. \] (No. 2.)

In applying this formula to Gay-Lussac's experiments, we have the constant quantity

\[ 2 \frac{2 \epsilon - \epsilon'}{g D} = r q = 647205 \]

for Gay-Lussac's 1st experiment. In order to find the height of the fluid in his 2nd tube by means of this constant quantity, we have

\[ r = \frac{190381}{2} = 0.951905, \text{ and } 2 \frac{2 \epsilon - \epsilon'}{g D} \times \frac{1}{3} = q = \frac{151311}{0.951095} \]

= 15.8956, from which, if we subtract one-sixth of the diameter, or 0.3173, we have 15.5783 for the altitude \( h \) of the lower point of the concavity of the meniscus, which differs only 0.0078 from 15.861 the observed altitude.

If we apply the same formula to Gay-Lussac's experiments on alcohol, we shall find the constant quantity

\[ 2 \frac{2 \epsilon - \epsilon'}{g D} = 6.0825 \text{ as deduced from the 1st experiment, and } h = 6.0725, \text{ which differs only 0.0100 from 6.08397, the altitude observed.} \]

From these comparisons, it is obvious that the mean altitudes, or the values of \( q \), are very nearly reciprocally proportional to the diameters of the tubes; for, in the experiments on water, the value of \( q \) deduced from this ratio is 15.895, which differs little from 15.9034, the value found from experiment; and that, in accurate experiments, the correction made by the addition of the sixth part of the diameter of the tube is indispensably requisite.

167. If the section of the pipe in which the fluid ascends is a rectangle, whose greater side is \( a \), and its lesser side \( d \), then the base of the elevated column will be \( a + d \), and its perimeter \( c = 2a + 2d \). Hence, the value of the meniscus will be

\[ \frac{ad^2}{2} - \frac{a\pi d^2}{8} = \frac{ad^2}{2} \left(1 - \frac{\pi}{4}\right), \]

that is

\[ q = h + \frac{d}{2} \left(1 - \frac{\pi}{4}\right). \]

Hence, if in the general equation No. 1, we substitute for \( c \) its equal \( 2a + 2d \), and for \( V \) its equal \( adq \), we have

\[ gDqad = 2 \frac{2 \epsilon - \epsilon'}{g D} \times 2a + 2d, \]

and dividing by \( a \) and by \( g D \), we have

\[ dq = 2 \frac{2 \epsilon - \epsilon'}{g D} \times 1 + \frac{d}{a}, \text{ and } q = 2 \frac{2 \epsilon - \epsilon'}{g D} \times 1 + \frac{d}{a}. \]

In applying this formula to the elevation of water between two glass plates, the side \( a \) is very great compared with \( d \), and therefore the quantity \( \frac{d}{a} \) being almost insensible, may be safely neglected. Hence the formula becomes

\[ q = 2 \frac{2 \epsilon - \epsilon'}{g D} \times \frac{1}{d}. \]

By comparing this formula with the formula No. 2, it is obvious that water will rise to the same height between plates of glass as in a tube, provided the distance \( d \) between the two plates of glass is equal to \( r \), or half the diameter of the tube. This result was obtained by Newton, and has been confirmed by the experiments of succeeding writers.

VOL. XII.

As the constant quantity \( 2 \frac{2 \epsilon - \epsilon'}{g D} \) is the same as already found for capillary tubes, we may take its value, viz. 15.1311, and substitute it in the preceding equation, we then have

\[ q = \frac{15.1311}{1.060} t = 14.1544; \text{ and since } \]

\[ h = q - \frac{d}{2} \left(1 - \frac{\pi}{2}\right), \text{ subtracting } \]

\[ \frac{d}{2} \left(1 - \frac{\pi}{4}\right) = 0.1147, \text{ we have } \]

\[ h = 14.0397, \text{ which differs very little from 13.574, the observed altitude.} \]

It will be seen from the formula No. 2, that of all tubes that have a prismatic form, the hollow cylinder is the one in which the volume of fluid raised is the least possible, as it has the smallest perimeter. It appears, also, that if the section of the tube is a regular polygon, the altitudes of the fluid will be reciprocally proportional to the homologous lines of the similar base, a result which, as we have seen, M. Gellert obtained from direct experiment. Hence, in all prismatic tubes whose sections are polygons inscribed in the same circle, the fluid will rise to the same mean height. If one of the two bases is, for example, a square, and the other an equilateral triangle, the altitudes will be as \( 2 : 3 \), or very nearly as \( 7 : 8 \).

168. M. La Place has remarked, that there may be several states of equilibrium in the same tube, provided its width is not uniform. If we suppose two capillary tubes communicating with one another, so that the smallest is placed above the greatest, we may then conceive their diameters and lengths to be such, that the fluid is at first in equilibrium above its level in the widest tube, and that in pouring in some of the same fluid, so as to reach the smaller tube, and fill part of it, the fluid will still maintain itself in equilibrium. When the diameter of a capillary tube diminishes by insensible gradations, the different states of equilibrium are alternately stable and instable. At first the fluid tends to raise itself in the tube, and this tendency diminishing, becomes nothing in a state of equilibrium. Beyond this it becomes negative, and consequently the fluid tends to descend. Thus the first equilibrium is stable, since the fluid, being a little removed from this state, tends to return to it. In continuing to raise the fluid, its tendency to descend diminishes, and becomes nothing in the second state of equilibrium. Beyond this it becomes positive, and the fluid tends to rise, and consequently to remove from this state which is not stable. In a similar manner it will be seen, that the third state is stable, the fourth instable, and so on.

169. Although the preceding method of considering the phenomena of capillary attraction is extremely simple and between accurate, yet it does not indicate the connexion which subsists between the elevation and depression of the fluid, and fluids and the concavity or convexity of the surface which every fluid assumes in capillary spaces. The object of M. La Place's first method, contained in his first supplement, is to determine this connexion.

By means of the methods for calculating the attraction of spheroids, he determines the action of a mass of fluid terminated by a spherical surface, concave or convex, upon a column of fluid contained in an infinitely narrow canal, directed towards the centre of this surface. By this action La Place means the pressure which the fluid contained in the canal would exercise, in virtue of the attraction of its entire mass upon a plane base situated in the interior of the canal, and perpendicular to its sides, at any sensible distance from the surface, this base being taken for unity. He then shews that this action is smaller when the surface Capillary is concave than when it is plane, and greater when the attraction, surface is convex. The analytical expression of this action &c. is composed of two terms. The first of these terms, which is much greater than the second, expresses the action of the mass terminated by a plane surface; and the second term expresses the part of the action due to the sphericity of the surface, or, in other words, the action of the meniscus comprehended between this surface and the plane which touches it. This action is either additive to the preceding, or subtractive from it, according as the surface is convex or concave. It is reciprocally proportional to the radius of the spherical surface; for the smaller that this radius is, the meniscus is the nearer to the point of contact.

From these results relative to bodies terminated by sensible segments of a spherical surface, La Place deduces this general theorem. "In all the laws which render the attraction insensible at sensible distances, the action of a body terminated by a curve surface upon an interior canal infinitely narrow, perpendicular to this surface in any point, is equal to the half sum of the actions upon the same canal of two spheres, which have for their radii the greatest and the smallest of the radii of the osculating circle of the surface at this point."

170. By means of this theorem, and the laws of hydrostatics, La Place has determined the figure which a mass of fluid ought to take when acted upon by gravity, or contained in a vessel of a given figure. The nature of the surface is expressed by an equation of partial differences of the second order, which cannot be integrated by any known method. If the figure of the surface is one of revolution, the equation is reduced to one of ordinary differences, and is capable of being integrated by approximation, when the surface is very small. La Place next shews, that a very narrow tube approaches the more to that of a spherical segment as the diameter of the tube becomes smaller. If these segments are similar in different tubes of the same substance, the radii of their surfaces will be inversely as the diameter of the tubes. This similarity of the spherical segments will appear evident, if we consider that the distance at which the action of the tube ceases to be sensible is imperceptible; so that if, by means of a very powerful microscope, this distance should be found equal to a millimetre, it is probable that the same magnifying power would give to the diameter of the tube an apparent diameter of several metres. The surface of the tube may therefore be considered as very nearly plane, in a radius equal to that of the sphere of sensible activity; the fluid in this interval will therefore descend, or rise from this surface, very nearly as if it were plane. Beyond this the fluid being subjected only to the action of gravity, and the mutual action of its own particles, the surface will be very nearly that of a spherical segment, of which the extreme planes being those of the fluid surface, at the limits of the sphere of the sensible activity of the tube, will be very nearly in different tubes equally inclined to their sides. Hence it follows that all the segments will be similar.

171. The approximation of these results gives the true cause of the ascent or descent of fluids in capillary tubes in the inverse ratio of their diameter. If in the axis of a glass tube we conceive a canal infinitely narrow, which bends round like the tube ABEDC in fig. 56, the action of the water in the tube in this narrow canal will be less, on account of the concavity of its surface, than the action of the water in the vessel on the same canal. The fluid will therefore rise in the tube to compensate for this difference of action; and as the concavity is inversely proportional to the diameter of the tube, the height of the fluid will be also inversely proportional to that diameter. If the surface of the interior fluid is convex, which is the case with mercury in a glass tube, the action of this fluid on the canal will be greater than that of the fluid in the vessel, and Capillary therefore the fluid will descend in the tube in the ratio of attraction, their difference, and consequently in the inverse ratio of the diameter of the tube.

In this manner of viewing the subject, the attraction of capillary tubes has no influence upon the ascent or depression of the fluids which they contain, but in determining the inclination of the first planes of the surface of the interior fluid extremely near the sides of the tube, and upon this inclination depends the concavity or convexity of the surface, and the length of its radius. The friction of the fluid against the sides of the tube may augment or diminish a little the curvature of its surface, of which we see frequent examples in the barometer. In this case the capillary effects will increase or diminish in the same ratio.

The differential equation of the surfaces of fluids enclosed in capillary spaces of revolution, conduits La Place to the following general result; that if into a cylindrical tube we introduce a cylinder which has the same axis as that of the tube, and which is such that the space comprehended between its surface and the interior surface of the tube has very little width, the fluid will rise in this space to the same height as in a tube whose radius is equal to this width. If we suppose the radii of the tube and of the cylinder infinite, we have the case of a tube included between two parallel and vertical planes, very near each other. This result has been confirmed, as we have already seen, by the experiments of Newton, Hauy, and Gay-Lussac. La Place then applies his theory to the phenomena presented by a drop of fluid, either in motion or suspended in equilibrium, either in a conical capillary tube, or between two plates, and inclined to each other, as discovered by Mr Hawksbee; to the mutual approximation of two parallel and vertical discs immersed in a fluid; to the phenomena which take place when two plates of glass are inclined to each other at a small angle; and to the determination of the figure of a large drop of mercury laid upon a horizontal plate of glass.

On the Form of Drops.

172. It was observed by M. Monge, that when drops of alcohol fall upon a surface of the same fluid, they do not at first mix with it, but roll over its surfaces with great facility, impinge against each other, and are reflected like billiard balls. M. Monge observed an analogous phenomenon in the drops of water which fall from the oars during the rowing of a boat, and during the condensation of the vapour of warm fluids.

In repeating the experiments of Monge, Sir David Brewster found that the phenomena were most beautiful when the capillary tube discharged the drops upon the inclined plane of fluid, which is elevated by the attraction of the edge of the cup. They ran down the inclined plane with great velocity, and sometimes even ascended the similar plane on the opposite side of the vessel. When the drop was discharged at the distance of one or two-tenths of an inch from the surface of the water, they had always the same magnitude when the tube was held in the same position; but when the point of the tube was brought within a tenth of an inch of the surface of the spirit of wine, this surface, instead of attracting the drop to it instantly, as Saussure would have predicted, actually resisted the gravity or weight of drop, and allowed it to attain a diameter nearly twice as great as it would have had, if it had been discharged in the ordinary manner. This swollen globe floated upon the surface in the same manner as the smaller drops, surrounded with a depression of the fluid surface similar to what is produced by a glass globe floating on mercury, or by the feet of particular insects, that have the power of running upon the surface of water. The floating Capillary globules are often produced even when they are discharged from a height of three or four inches; and by letting them fall upon the inclined plane of fluid formerly mentioned, they will often rebound from the surface, and fall over the sides of the cup.

173. In a phenomenon the very reverse of the formation of a drop, which was first noticed by Sir David Brewster, the cohesion of fluids is shown in a very interesting manner. If we take a phial, with a wide mouth, half filled with Canada balsam, and allow the balsam to flow to the mouth of the phial and fill it up, then when the phial is placed on its bottom, a fine transparent film of balsam will be seen extending over the mouth of the phial. If we now take a piece of slender wire, and touch the film near the middle, so as to tear away a little part of it, the remaining part of the film which has been elevated by this force will descend to its level position, and the ragged aperture from which the balsam has been torn will be seen to assume a form perfectly circular, having its edge in a slight degree thickened, like a circle with a raised margin turned out of a piece of wood. This fine circular aperture grows wider and wider, and continues to preserve its circular form till the mouth of the phial is again opened.

174. We shall now conclude the subject of capillary attraction with an account of an experiment made by Sir David Brewster, and intimately connected with the subject. Above a vessel MNOP, Fig. 57, nearly filled with water, a convex lens LL was placed at the distance of the 10th of an inch, and rays R, R, R, were incident upon its upper surface. The focus of these rays was at F, a little beyond the bottom of the vessel, so that a circular image of the luminous object was seen on the bottom of the vessel, having AB for its diameter. If the lens is now made to descend gradually towards the surface of the water, and the eye kept steadily upon the luminous image AB, a dark spot will be seen at D in the centre of AB, a little while before the lens attracts Capillary and elevates the water MN. Sometimes this spot may Attraction, be seen playing backwards and forwards by the slight motion of the hand, so that the lens can even be withdrawn from the fluid surface without having actually touched it. In general, however, the sudden rise of the water to the lens follows the appearance of the blackspot. When the water is in contact with the glass, the focus of the rays R, R, is now transferred to f, and the circular image on the bottom is now ab, and the intensity of the light in this circle is to that in the circle AB, as \( \frac{AB^2}{ab^2} \). Now it is obvious, that the darkish spot at D is just the commencement of the transference of the focus from F to f; or when the dark spot is produced, the progress of the rays is the same as if the focus were transferred to f. This remarkable effect may arise from two causes. 1. The approach of the lens to the surface MN, may occasion a depression mon in the surface of the fluid of the same curvature as L/L, which would have the effect of transferring the focus from F to f. 2. The transference of the focus from F to f may arise from the optical contact of the glass of water taking place at a greater distance from the lens than that at which capillary attraction commences.

PART II.—HYDRAULICS.

175. Hydraulics is that branch of the science of Hydrodynamics which relates to fluids in motion. It comprehends the theory of running water, whether issuing from orifices in reservoirs by the pressure of the superincumbent mass, or rising perpendicularly in jets d'eau from the pressure of the atmosphere; whether moving in pipes and canals, or rolling in the beds of rivers. It comprehends also the resistance or the percussion of fluids, and the oscillation of waves.

CHAPTER I.—THEORY OF FLUIDS ISSUING FROM ORIFICES IN RESERVOIRS, EITHER IN A LATERAL OR A VERTICAL DIRECTION.

176. If water issues from an orifice either in the bottom or side of a reservoir, the surface of the fluid in the reservoir is always horizontal till it reaches within a little of the bottom. When a vessel, therefore, is emptying itself, the particles of the fluid descend in vertical lines, as is represented in fig. 58.; but when they have reached within three or four inches of the orifice mn, the particles which are not immediately above it change the direction of their motion, and make for the orifice in directions of different degrees of obliquity. The velocities of these particles may be decomposed into two others, one in a horizontal direction, by which they move parallel to the orifice, and the other in a vertical direction, by which they approach that orifice. Now, as the particles about C and D move with greater obliquity than those nearer E, their horizontal velocities must also be greater, and their vertical velocities less. But the particles near E move with so little obliquity that their cause of vertical are much greater than their horizontal velocities, and the very little less than their absolute ones. The different particles of the fluid, therefore, will rush through the orifice mn with very different velocities, and in various directions, and will arrive at a certain distance from the orifice in different times. On account of the mutual adhesion of the fluid particles, however, those which have the greatest velocity drag the rest along with them; and as the former move through the centre of the orifice, the breadth of the issuing column of fluid will be less at op than the width of the orifice mn.

177. That the preceding phenomena really exist when a vessel of water is discharging its contents through an aperture, experience sufficiently testifies. If some small substances specifically heavier than water be thrown into the fluid when the vessel is emptying itself, they will at first descend vertically, and when they come within a few inches of the bottom they will deviate from this direction, and describe oblique curves similar to those in the figure. The contraction of the vein or column of fluid at \( op \) is also manifest from observation. It was first discovered by Sir Isaac Newton, and denominated the *vena contracta*. The greatest contraction takes place at a point \( o \), whose distance from the orifice is equal to half its diameter, so that

\[ om = \frac{mn}{2} \]

and the breadth of the vein or column of fluid at \( o \) is to the width of the orifice as 5 to 8 according to Bossut, or as 5.197 to 8 according to the experiments of Michelotti, the orifice being perforated in a thin plate. But when the water is made to issue through a short cylindrical tube, the same contraction, though not obvious to the eye, is so considerable, that the diameter of the contracted vein is to that of the orifice as 6.5 to 8.

If \( A \) therefore be the real size of the orifice in a thin plate, its corrected size, or the breadth of the contracted vein, will be

\[ \frac{5.197 \times A}{8} \]

and when a cylindrical tube is employed it will be

\[ \frac{13 \times A}{16} \]

In the first case the height of the water in the reservoir must be reckoned from the surface of the fluid to the point \( o \), where the vein ceases to contract; and when a cylindrical tube is employed, it must be reckoned from the same surface to the exterior aperture of the tube.

178. Suppose the fluid ABCD (fig. 58) divided into an infinite number of equal strata or laminae by the horizontal surfaces MN, gh infinitely near each other; and let \( mnop \) be a small column of fluid which issues from the orifice in the same time that the surface MN descends to gh. The column \( mnop \) is evidently equal to the lamina MN gh, for the quantity of fluid which is discharged during the time that MN descends to gh, is evidently MN gh; and to the quantity discharged in that time, the column \( mnop \) was equal by hypothesis. Let A be the area of the base MN, and B the area of the base \( mn \); let \( x \) be the height of a column equal to MN gh, and having A for its base, and let \( y \) be the height of the column \( mnop \). Then, since the column \( mnop \) is equal to the lamina MN gh, we shall have

\[ Ax = By \]

and (Geometry, Sect. IV. Theor. IX.) \( x : y = A : B \); but as the surface MN descends to gh in the same time that \( mn \) descends to \( op \), \( x \) will represent the mean velocity of the lamina MN gh, and \( y \) the mean velocity of the column \( mnop \). The preceding analogy, therefore, informs us, that the mean velocity of any lamina is to the velocity of the fluid issuing from the orifice reciprocally as the area of the orifice is to the area of the base of the lamina MN gh. Hence it follows, that if the area of the orifice is infinitely small, with regard to the area of the base of the lamina into which the fluid is supposed to be divided, the mean velocity of the fluid at the orifice will be infinitely greater than that of the lamina; that is, while the velocity at the orifice is finite, that of the lamina will be infinitely small.

179. Before applying these principles to the theory of hydraulics, it may be proper to observe, that several distinguished philosophers have founded the science upon the same general law from which we have deduced the principles of hydrostatics (48). In this way they have represented the motion of fluids in general formulae; but these formulae are so complicated from the very nature of the theory, and the calculations are so intricate, and sometimes impracticable from their length, that they can afford no assistance to the practical engineer.

**Definition.**

180. If the water issues at \( mn \) with the same velocity \( V \) that a heavy body would acquire by falling freely through a given height \( H \), this velocity is said to be due to the height \( H \), and inversely the height \( H \) is said to be due to the velocity \( V \).

**Prop. I.**

181. The velocity of a fluid issuing from an infinitely small orifice in the bottom or side of a vessel, is equal to that which is due to the height of the surface of the fluid above that orifice, the vessel being supposed constantly full.

Let AB, fig. 59, be the vessel containing the fluid, its velocity when issuing from the aperture \( mn \) will be that which is due to the height \( Dm \), or equal to that which a heavy body would acquire by falling through that height. Because the orifice \( mn \) is infinitely small, the velocity of the laminae into which the fluid may be supposed to be divided, will also be infinitely small. But since all the fluid particles, by virtue of their gravity, have a tendency to descend with the same velocity; and since the different laminae of the fluid lose this velocity, the column \( mnst \) must be pressed by the superincumbent column \( Dmn \); and calling S the specific gravity of the fluid, the moving force which pushes out the column \( mnst \) will be \( S \times Dm \times mn \) (art. 58). Now, let us suppose, that, when this moving force is pushing out the column \( mnst \), the absolute weight of the column \( mnop \), which may be represented by \( S \times mn \times np \), causes itself to fall through the height \( np \). Then, if \( V, U \) be the velocities impressed upon the columns \( mnst \), and \( mnop \), by the moving forces \( S \times Dm \times mn \), and \( S \times mn \times np \); these moving forces must be proportional to their effects, or to the quantities of motion which they produce, that is, to \( V \times mnst \) and \( U \times mnop \), because the quantity of motion is equal to the velocity and mass conjointly; hence we shall have

\[ S \times Dm \times mn : S \times mn \times np = V : mnst : U : mnop \]

But since the volumes \( mnst, mnop \) are to one another as their heights \( mo, os \), and as these heights are run through in equal times, and consequently represent the velocity of their motion, \( mnst \) may be represented by \( V \times mn \) and \( mnop \) by \( U \times mn \); therefore we shall have

\[ S \times Dm \times mn : S \times mn \times np = V \times mn : U \times mn \]

and dividing by \( mn \),

\[ S \times Dm : np = V^2 : U^2 \]

Now let \( v \) be the velocity due to the height \( Dm \), then (see Mechanics) \( np : U^2 = Dm : v^2 \); but since \( S \times Dm : S \times np = V^2 : U^2 \); then by (Euclid V. 15), and by permutation \( Dm : V^2 = np : U^2 \); therefore by substitution (Euclid V. 11), \( Dm : V^2 = Dm : v^2 \), and (Euclid V. 9), \( V^2 = v^2 \) or \( V = v \). But \( V \) is the velocity with which the fluid issues from the orifice \( mn \), and \( v \) the velocity due to the height \( Dm \); therefore, since the velocities are equal, the proposition is demonstrated.

182. Cor. 1. If the vessel AB empties itself by the small orifice \( mn \), so that the surface of the fluid takes successively the positions OP, QR, ST, the velocities with which the water will issue when the surfaces have these positions will be those due to the heights \( En, Fn, Gn \), for in these different positions the moving forces are the columns \( Emn, Fmn, Gmn \).

183. Cor. 2. Since the velocities of the issuing fluid when its surface is at \( E_n, F_n, G_n \) are those due to the heights \( E_n, F_n, G_n \), it follows from the properties of falling bodies (see Mechanics), that if these velocities were continued uniformly, the fluid would run through spaces equal to \( 2E_n, 2F_n, 2G_n \) respectively, in the same time that a heavy body would fall through \( E_n, F_n, G_n \), respectively.

184. Cor. 3. As fluids press equally in all directions, of the preceding proposition will hold true, when the orifices are at the sides of vessels, and when they are formed to throw the fluid upwards, either in a vertical or an inclined direction, provided that the orifices are in these several cases at an equal distance from the upper surface of the fluid. This corollary holds also in the case mentioned in Cor. 1.

185. Cor. 4. When the fluid issues vertically, it will rise to a height equal to the perpendicular distance of the orifice from the surface of the fluid; (see Mechanics), this is true of falling bodies in general, and must therefore be true in the case of water; owing to the resistance of the air, however, and the friction of the issuing fluid upon the sides of the orifice, jets of water do not exactly rise to this height.

186. Cor. 5. As the velocities of falling bodies are as the square roots of the heights through which they fall (see Mechanics), the velocity \( V \) of the effluent water when the surface is at \( E_2 \) will be to its velocity \( v \) when the surface is at \( G \), as \( \sqrt{E_2} : \sqrt{G} \). (Cor. 1.) That is, the velocities of fluids issuing from a very small orifice are as the square roots of the altitude of the water above these orifices. As the quantities of fluids discharged are as the velocities, they will also be as the square roots of the altitude of the fluid. This corollary holds true of fluids of different specific gravities, notwithstanding Belidor (Architect. Hydraul., tom. i. p. 187), has maintained the contrary; for though a column of mercury \( D \) presses with 14 times the force of a similar column of water, yet the column \( mnp \) (fig. 59) of mercury which is pushed out is also 14 times as heavy as a similar column of water; and as the resistance bears the same proportion to the moving force, the velocities must be equal.

187. Cor. 6. When a vessel is emptying itself, if the area of the laminae into which we may suppose it divided, be everywhere the same, the velocity with which the surface of the fluid descends, and also the velocity of efflux, will be uniformly retarded. For as the velocity \( V \) with which the surface descends is to the velocity \( v \) at the orifice, as the area \( a \) of the orifice to the area \( A \) of the surface, then \( V : v = a : A \); but the ratio of \( a : A \) is constant, therefore \( V \) varies as \( v \), that is, \( V : V' = v : v' \); but, (Cor. 1.) \( v : v' = \sqrt{h} : \sqrt{h'} \), \( h \) being the height of the surface above the orifice, therefore \( V : V' = \sqrt{h} : \sqrt{h'} \). But this is the property of a body projected vertically from the earth's surface, and as the retarding force is uniform in the one case (see Mechanics), it must also be uniform in the other.

188. Cor. 7. If a cylindrical vessel be kept constantly full, twice the quantity contained in the vessel will run out during the time in which the vessel would have emptied itself. For (Cor. 2 and 6) the space through which the surface of the fluid at \( D \) would descend if its velocity continued uniform being \( 2Dm \), double of \( Dm \) the space which it actually describes in the time it empties itself; the quantity discharged in the former case will also be double the quantity discharged in the latter; because the quantity discharged when the vessel is kept full, may be measured by what the descent of the surface would be, if it could descend with its first velocity.

Scholium.

189. The reader will probably be surprised when he finds in some of our elementary works on hydrostatics, Motion of that the velocity of the water at the orifice is only equal to Fluids, &c., that which a heavy body would acquire by falling through half the height of the fluid above the orifice. This was first maintained by Sir Isaac Newton, who found that the diameter of the vena contracta was to that of the orifice as 21 to 25. The area therefore of the one was to the area of the other as \( 21^2 : 25^2 \), which is nearly the ratio of 1 to \( \sqrt{2} \). But by measuring the quantity of water discharged in a given time, and also the area of the vena contracta, Sir Isaac found that the velocity at the vena contracta was that which was due to the whole altitude of the fluid above the orifice. He therefore concluded, that since the velocity at the orifice was to that at the vena contracta as 1 : \( \sqrt{2} \), and in the latter velocity was that which was due to the whole altitude of the fluid, the former velocity, or that at the orifice, must be that which is due to only half that altitude, the velocities being as the square roots of the heights. Now the difference between this theory and that contained in the preceding proposition may be thus explained. The velocity found by the preceding proposition is evidently the vertical velocity of the filaments at \( E \) (fig. 59), which being immediately above the centre of the aperture \( mn \) are not diverted from their course, and have therefore their vertical equal to their absolute velocity. But the vertical velocity of the particles between \( C \) and \( E \), and \( E \) and \( D \), is much less than their absolute velocity, on account of the obliquity of their motion, and also on account of their friction on the sides of the orifice. The mean vertical velocity, consequently, of the issuing fluid will be much less than the vertical velocity of the particles at \( E \), that is, than the velocity found by the above proposition, or that due to the height \( Dm \). Now the velocity found by Sir Isaac Newton from measuring the quantity of water discharged, was evidently the mean velocity, which ought to be less than the velocity given by the preceding proposition, the two velocities being as \( 1 : \sqrt{2} \), or as \( 1 : 1.414 \). The theorem of Newton therefore may be considered as giving the mean velocity at the orifice, while the proposition gives the velocity of the particles at \( D \), or the velocity at the vena contracta.

Prop. II.

190. To find the quantity of water discharged from a very small orifice in the side or bottom of a reservoir, the time of discharge, and the altitude of the fluid, the vessel being kept constantly full, and any two of these quantities being given.

Let \( A \) be the area of the orifice \( mn \); \( W \) the quantity of water discharged in the time \( T \); \( H \) the constant height \( Dm \) of the water in the vessel, and let 16.087 feet be the height through which a heavy body descends in a second of time. Now, as the times of description are proportional to the square roots of the heights described, the time in which a heavy body will fall through the height \( H \), will be found from the following analogy: \( \sqrt{16.087} : \sqrt{H} = 1 : \frac{\sqrt{H}}{16.087} \), the time required. But as the velocity at the orifice is uniform, a column of fluid whose base is \( mn \) and altitude \( 2H \) (Prop. I. Cor. 2.) will issue in the time 16.087 \( \sqrt{H} \), or since \( A \) is the area of the orifice \( mn \), \( A \times 2H \) or

---

1 When a fluid runs through a conical tube kept constantly full, the velocities of the fluid in different sections will be inversely as the area of the sections. For as the same quantity of fluid runs through every section in the same time, it is evident that the velocity must be greater in a smaller section, and as much greater as the section is smaller, otherwise the same quantity of water would not pass through each section in the same time. Now the area of the vena contracta is to the area of the orifice, as \( 1 : \sqrt{2} \), therefore the velocity at the vena contracta must be to the velocity at the orifice as \( \sqrt{2} : 1 \). Motion of 2HA will represent the column of fluid discharged in that Fluids, &c., time. Now since the quantities of fluid discharged in different times must be as the times of discharge, the velocity at the orifice being always the same, we shall have \( \frac{\sqrt{H}}{16.087} : T = 2HA : W \), and (Geometry, Sect. IV. Theor. VIII.)

\[ W \sqrt{H} = 2HAT \text{ or } W = \frac{2HAT \times 16.087}{\sqrt{H}} \]

and since

\[ \frac{H}{\sqrt{H}} = \sqrt{H} \text{ we shall have } W = 2AT \sqrt{H} \times 16.087, \]

an equation from which we deduce the following formulae, which determine the quantity of water discharged, the time of discharge, the altitude of the fluid, and the area of the orifice, any three of these four quantities being given:

\[ W = 2AT \sqrt{H} \times 16.087 \quad A = \frac{W}{2T \sqrt{H} \times 16.087} \]

\[ H = \frac{W^2}{4A^2T^2 \times 16.087} \quad T = \frac{W}{2A \sqrt{H} \times 16.087} \]

191. It is supposed in the preceding proposition that the orifice in the side of the vessel is so small, that every part of it is equally distant from the surface of the fluid. But when the orifice is large like M (fig. 60.), the depths of different parts of the orifice below the surface of the fluid are very different, and consequently the preceding formulae will not give very accurate results. If we suppose the orifice M divided into a number of smaller orifices \( a, b, c \), it is evident that the water will issue at \( a \) with a velocity due to the height \( Da \), the water at \( b \) with a velocity due to the height \( Eb \), and the water at \( c \), with a velocity due to the height \( Ec \). When the whole orifice, therefore, is opened, the fluid will issue with different velocities at different parts of its section. Consequently, in order to find new formulae expressing the quantity of water discharged, we must conceive the orifice to be divided into an infinite number of areas or portions by horizontal planes; and by considering each area as an orifice, and finding the quantity which it will discharge in a given time, the sum of all these quantities will be the quantity discharged by the whole orifice M.

**Prop. III.**

192. To find the quantity of water discharged by a rectangular orifice in the side of a vessel kept constantly full.

Let ABD (fig. 61.) be the vessel with the rectangular orifice GL, and let AB be the surface of the fluid. Draw the lines MNOP, m nop, infinitely near each other, and from any point D draw the perpendicular DC meeting the surface of the fluid in C. Then regarding the infinitely small rectangle MO mo as an orifice whose depth below the surface of the fluid is H, we shall have by the first of the preceding formulae, the quantity of water discharged in the time T, or \( W = \frac{T \sqrt{16.087} \times \sqrt{CN} \times 2MO \times Nn}{CN} \), CN being equal to H and MO \(\times\) Nn to the area A. As the preceding formula represents the quantity of fluid discharged by each elementary rectangular orifice, into which the whole orifice GL is supposed to be divided, we must find the sum of all the quantities discharged in the time T, in order to have the total quantity afforded by the finite orifice in the same time. Upon DC as the principal axis, describe the parabola BCA, having its parameter P equal to 4DC. Continue FG and DK to H and E. The area NP pn may be expressed by NP \(\times\) Nn. But (Conic Sections, Part I. Prop. X.) \(NP^2 = CN \times P\) (P being the parameter of the parabola), therefore NP = \(\sqrt{CN} \times P\), and multiplying by Nn we have NP \(\times\) Nn = Nn \(\times\) CN \(\times\) P, which expresses the area NP pn. Now this expression of the elementary area being multiplied by the constant quantity

\[ T \sqrt{16.087} \times \frac{MO}{\sqrt{P}} \]

gives for a product \( T \sqrt{16.087} \times \frac{MO}{\sqrt{P}} \times \sqrt{CN} \times 2MO \times Nn \), for \( \frac{\sqrt{P}}{P} = \frac{1}{2} \sqrt{P} \) and \( \frac{MO \times \sqrt{P}}{\sqrt{P}} = 2MO \).

But this product is the very same formula which expresses the quantity of water discharged in the time T by the orifice MO om. Therefore, since the elementary area MP pm multiplied by the constant quantity

\[ T \sqrt{16.087} \times \frac{MO}{\sqrt{P}} \]

gives the quantity of water discharged by the orifice MO om in a given time, and since the same may be proved of every other orifice of the same kind into which the whole orifice is supposed to be divided, we may conclude that the quantity of water discharged by the whole orifice GL will be found by multiplying the parabolic area FHED by the same constant quantity \( T \sqrt{16.087} \times \frac{MO}{\sqrt{P}} \).

Now the area FHED is equal to the difference between the areas CDE and CFH. But (Conic Sections, Part I. Prop. X.) the area CDE = \( \frac{3}{4} CD \times DE \); and since \( P = 4CD \), and (Conic Sections, Part I. Prop. X.) \( DE^2 = CD \times P \)

we have \( DE^2 = CD \times 4CD = 4CD^2 \), that is \( DE = 2CD \),

then by substituting this value of DE in the expression of the area CDE, we have \( CDE = \frac{3}{4} CD^2 \). The area CFH = \( \frac{3}{4} CF \times FH \), consequently the area FHED = \( \frac{3}{4} CD^2 - \frac{3}{4} CF \times FH \), which, multiplied by the constant quantity, gives for the quantity of water discharged, (\( \frac{3}{4} P^2 \) being substituted instead of its equal \( \frac{3}{4} CD^2 \)),

\[ W = \frac{T \sqrt{16.087} \times MO \times \frac{1}{2} P^2 - \frac{3}{4} CF \times FH}{\sqrt{P}}. \]

But by the property of the parabola \( FH^2 = CF \times P \) and \( FH = \sqrt{CF \times P} \), therefore substituting this value of FH in the preceding formula, and also \( \frac{1}{2} \sqrt{P} \) for its equal \( \frac{\sqrt{P}}{P} \) we have

\[ W = \frac{T \sqrt{16.087} \times MO \times \frac{1}{2} P^2 - \frac{3}{4} CF \times \sqrt{CF}}{\sqrt{P}} \]

and dividing by \( \frac{1}{2} \sqrt{P} \) gives us

\[ W = \frac{T \sqrt{16.087} \times MO \times \frac{1}{2} P \sqrt{P} - \frac{3}{4} CF \times \sqrt{CF}}{\sqrt{P}} \]

hence

\[ T = \frac{W}{\sqrt{16.087} \times MO \times \frac{1}{2} P \sqrt{P} - \frac{3}{4} CF \times \sqrt{CF}} \]

\[ MO = \frac{W}{T \sqrt{16.087} \times \frac{1}{2} P \sqrt{P} - \frac{3}{4} CF \times \sqrt{CF}} \]

\[ P = \frac{9W}{4T \sqrt{16.087} + 3CF \sqrt{CF}} \] In these formulas W represents the quantity of water discharged, T the time of discharge, MO the horizontal width of the rectangular orifice, P the parameter of the parabola = 4CD, CD the depth of the water in the vessel or the altitude of the water above the bottom of the orifice, and CF the altitude of the water above the top of the orifice. The vertical breadth of the orifice is equal to \( \frac{CD - CF}{2} \).

193. Let \( x \) be the mean height of the fluid above the orifice, or the height due to a velocity, which, if communicated to all the particles of the issuing fluid, would make the same quantity of water issue in the time \( T' \), as if all the particles moved with the different velocities due to their different depths below the surface, then by Prop. II., the quantity discharged or \( W = 2T \times MO \times CD - CF \times \sqrt{x \times 16.087} \); the area of the orifice being \( MO \times CD - CF \), and by making this value of \( W \) equal to its value in the preceding article, we have the following equation:

\[ 2T \times MO \times CD - CF \times \sqrt{x \times 16.087} = T \times \sqrt{16.087} \]

\[ \times MO \times \frac{P}{2} \times \frac{CF}{2} \times \sqrt{CF}, \text{ which, by division and reduction, and the substitution of } \frac{P}{2} \text{ instead of } CD \text{ its equal, becomes} \]

\[ x = \frac{\frac{P}{2} \times \frac{CF}{2}}{\frac{P}{2} + \frac{CF}{2}}. \]

Now this value of \( x \) is evidently different from the distance of the centre of gravity of the orifice from the surface of the fluid, for this distance is \( \frac{CD + CF}{2} \) or \( \frac{P + CF}{2} \). But in proportion as CE increases, the other quantities remaining the same, the value of \( x \) will approach nearer the distance of the centre of gravity of the orifice from the surface of the fluid; for when CF becomes infinite, the parabolic arch CHE will become a straight line, and consequently the mean ordinate of the curve, which is represented by the mean velocity of the water, will pass through the middle of FD or the centre of gravity of the orifice.

**Prop. IV.**

194. To find the time in which a quantity of fluid equal to ABRT, will issue out of a small orifice in the side or bottom of the vessel AB, that is, the time in which the surface AB will descend to RT.

Draw DE, de at an infinitely small distance, and parallel to AB. The lamina of fluid DdeE may be represented by \( DE \times ob \); DE expressing the area of the surface. When the surface of the water has descended to DE, the quantity of fluid which will be discharged by an uniform motion of velocity in the time \( T \), will be \( T \times \frac{16.087}{2A} \times \sqrt{om} \), Fluids, &c. A being the area of the orifice, as in Prop. II. But as the variation in the velocity of the water will be infinitely small, when the surface descends from DE to de, its velocity may be regarded as uniform. The time, therefore, in which the surface describes the small height ob will be found by the following analogy; \( T \times \frac{16.087}{2A} \times \sqrt{om} : \)

\[ T = DE \times ob : \frac{DE \times ob}{\sqrt{16.087} \times 2A \times \sqrt{om}}. \]

Now as this formula expresses the time in which the surface descends from DE to de, and as the same may be shewn of every other elementary portion of the height CS, the sum of all these elementary times will give us the value of \( T \), the time in which the surface AB falls down to RT. For this purpose draw GP equal and parallel to CM, and upon it as an axis, describe the parabola PVQ, having its parameter P equal to 4GP. Continue the lines AB, DE, de, RT, so as to form the ordinates HF, hF, UV, of the parabola. Upon GP as an axis, describe a second curve, so that the ordinate GM may be equal to the area of the surface at AB, divided by the corresponding ordinate GQ of the parabola, and that the ordinate HR may be the quotient of the area of the surface at DE divided by the ordinate HF. Now (Conic Sections, Part I. Prop. X.) \( HF^2 = HP \times P \), or \( HF = \sqrt{HP \times P} \), that is \( \sqrt{HP} = \frac{HF}{\sqrt{P}} \); and since \( om = HP ; \frac{DE}{\sqrt{om}} = \frac{DE \times \sqrt{P}}{HF} \). But by the construction of the curve MN, we have \( \frac{DE}{\sqrt{om}} = HR \times \sqrt{P} \). The elementary time, therefore, expressed by \( \frac{DE \times ob}{\sqrt{16.087} \times 2A \times \sqrt{om}} \) will, by the different substitutions now mentioned, be \( \frac{HR \times ob \times \sqrt{P}}{2A \times \sqrt{16.087}} \) or \( \frac{\sqrt{P}}{2A \times \sqrt{16.087}} \times HR \times ob \). But the factor \( \frac{\sqrt{P}}{2A \times \sqrt{16.087}} \) consisting of constant quantities is itself constant, and the other factor \( HR \times ob \) represents the variable curvilinear area HRsh. Now as the same may be shewn of every other element of the time \( T \), compared with the corresponding elements of the area GUTM, it follows that the time \( T \) required, will be found by multiplying the constant quantity \( \frac{\sqrt{P}}{2A \times \sqrt{16.087}} \) by the curvilinear area GUTM; therefore \( T = \frac{\sqrt{P}}{\sqrt{16.087}} \times \frac{GUTM}{2A} \), and the time in which the surface descends to \( ob \), or in which the vessel empties itself, will be equal to \( \frac{\sqrt{P}}{\sqrt{16.087}} \times \frac{GPNM}{2A} \).

Cor. The quantity of fluid discharged in the given time \( T \) may be found by measuring the contents of the vessel AB between the planes AB, and RT; the descent of the surface AB, viz. the depth CS, being known.

**Prop. V.**

195. To find the time in which a quantity of fluid equal to ABRT will issue out of a small orifice in the side or bottom of the cylindrical vessel AB, that is, the time in which the surface AB will descend to RT.

Let us suppose that a body ascends through the height mC (fig. 63.) with a velocity increasing in the same manner as if the vessel AB were inverted, and the body fell from m to C. The velocity of the ascending body at different points of its path being proportional to the square roots of the heights described, will be expressed by the ordinates of the parabola PVQ. The line DE being infinitely near to de, as soon as the body arrives at b it will describe the small space bo or hH in a portion of time infinitely small, with a velocity represented by the ordinate HF. Now the time in which the body will ascend through the space mC or its equal PG will be \( \frac{\sqrt{PG}}{\sqrt{16.087}} \), because

\[ \sqrt{16.087} : \sqrt{PG} = \sqrt{PG} : \sqrt{16.087} \quad \text{(see Mechanics);} \]

and if the velocity impressed upon the body when at C were continued uniformly, it would run through a space equal to 2GP or GQ in the time \( \frac{\sqrt{PG}}{\sqrt{16.087}} \). But (Dynamics, 22.) the times of description are as the spaces described directly, and the velocities inversely, and therefore the time of describing the space 2GP or GQ uniformly, viz. the time \( \frac{\sqrt{PG}}{\sqrt{16.087}} \) will be to the time of describing the space hH uniformly as \( \frac{GQ}{GQ} : \frac{Hh}{HF} \), that is, as \( \frac{GQ}{GQ} \) or \( 1 : \frac{\sqrt{PG}}{\sqrt{16.087}} \).

\( Hh : \frac{\sqrt{PG}}{\sqrt{16.087}} \times \frac{Hh}{HF} \) the time in which the ascending body will describe Hh uniformly; but PG being equal to \( \frac{1}{4}P \), the parameter of the parabola, we shall have \( \sqrt{PG} = \sqrt{\frac{1}{4}P} = \frac{1}{2}\sqrt{P} \). Substituting this value of \( \sqrt{PG} \) in the last formula, we shall have for the expression of the time of describing Hh uniformly \( \frac{\sqrt{P}}{\sqrt{16.087}} \times \frac{Hh}{HF} \). But by Prop. IV. the time in which the surface DH descends into the position dh, that is, in which it describes Hh is represented by \( \frac{\sqrt{P}}{2A\sqrt{16.087}} \times Hr \times ob \) or \( \frac{\sqrt{P}}{\sqrt{16.087}} \times \frac{Hr \times Hh}{2A} \).

Therefore the time in which the ascending body moves through Hh, is to the time in which the descending surface moves through Hh as \( \frac{\sqrt{P}}{\sqrt{16.087}} \times \frac{Hh}{HF} : \frac{\sqrt{P}}{\sqrt{16.087}} \times \frac{Hr \times Hh}{2A} \), which expressions, after being multiplied by 2, and after substituting in the latter \( \frac{DE}{HF} \) instead of \( Hr \), which is equal to it by construction, will become \( \frac{\sqrt{P}}{\sqrt{16.087}} \times \frac{Hh}{HF} : \frac{\sqrt{P}}{\sqrt{16.087}} \times \frac{DE \times Hh}{A \times HF} \), DE representing, in this and in the following proposition, the area of the surface of the fluid at D. Now, if we multiply the first of these expressions by DE, and the second by A, we shall find the two products equal; consequently (Euclid VI. 16.), the first expression is to the second, or the time of the body's ascent through Hh is to the time of the surface's descent through Hh, as the area A of the orifice is to the area DE of the base of the cylindrical vessel; and as the same may be demonstrated of every elementary time in which the ascending body and the descending surface describe equal spaces, it follows that the whole time in which the ascending body will describe the height mC or PG, is to the whole time in which the surface AB will descend to mn, or in which the vessel will empty itself, as the area A of the orifice is to the area of the surface DE, that is, \( A : DE = \frac{\sqrt{PG}}{\sqrt{16.087}} : \frac{\sqrt{PG}}{\sqrt{16.087}} \times \frac{DE}{A} \), the time in which the vessel AB will empty itself. If RTmn be the vessel, it may be shewn in the same manner, that the time in which it will empty itself will be \( \frac{\sqrt{PU}}{\sqrt{16.087}} \times \frac{DE}{A} \), DE being equal to RT. But the difference between the times in which the vessel ABmn empties itself, and the time in which the vessel RTmn empties itself, will be equal to the time required in the proposition, during which the surface AB descends to RT. This time therefore will be

\[ T = \frac{\sqrt{PG}}{\sqrt{16.087}} \times \frac{DE}{A} \times \frac{\sqrt{PU}}{\sqrt{16.087}} \times \frac{DE}{A} = \frac{DE \times \sqrt{PG} - DE \times \sqrt{PU}}{A \sqrt{16.087}} \]

\[ T = \frac{DE \times \sqrt{PG} - \sqrt{PU}}{A \sqrt{16.087}}. \quad \text{Hence} \]

\[ PU = \left( \frac{T, A \sqrt{16.087} - \sqrt{PG}}{DE} \right)^2 \]

\[ PG = \left( \frac{T, A \sqrt{16.087} + \sqrt{PU}}{DE} \right)^2 \]

\[ PG - PU \text{ or } UG = 2T, A \times DE \times \sqrt{PG} \times \frac{16.087}{DE^2} - T^2 A^2 \times \frac{16.087}{DE^2} \]

As the quantity of fluid discharged while the surface AB descends to RT is equal to DE × UG, we shall have

\[ W = DE \times 2T, A \times DE \times \sqrt{PG} \times \frac{16.087}{DE^2} - T^2 A^2 \times \frac{16.087}{DE^2} \]

\[ A = \frac{DE \times \sqrt{PG} \times \sqrt{PU}}{T \sqrt{16.087}} \]

\[ DE = \frac{T, A \sqrt{16.087}}{\sqrt{PG} - \sqrt{PU}} \] Prop. VI.

196. If two cylindrical vessels are filled with water, the time in which their surfaces will descend through similar heights will be in the compound ratio of their bases, and the difference between the square roots of the altitudes of each surface at the beginning and end of its motion, directly, and the area of the orifices inversely.

Let \( AB \) \( mn \), \( A'B' \) \( m'n' \) be the two vessels; then by the last proposition, the time \( T \), in which the surface \( AB \) of the first descends to \( DE \) and \( RT \), will be to the time \( T' \) in which the surface \( A'B' \) of the second descends to \( R'T' \) as

\[ \frac{DE \times \sqrt{PG} \times \sqrt{PU}}{A \times \sqrt{16.087}} \]

to

\[ \frac{D'E' \times \sqrt{PG'} - \sqrt{PU'}}{A' \times \sqrt{16.087}}, \]

or by dividing by \( \sqrt{16.087} \), as

\[ \frac{DE \times \sqrt{PG} - \sqrt{PU}}{A}. \]

Q.E.D.

Hours.

Distance of each Hour above the bottom.

Number of Parts in each Hour.

| Hours | Distance | Number | |-------|----------|--------| | 0 | 144 | 23 | | 1 | 121 | 21 | | 2 | 100 | 19 | | 3 | 81 | 17 | | 4 | 64 | 15 | | 5 | 49 | 13 | | 6 | 36 | 11 | | 7 | 25 | 9 | | 8 | 16 | 7 | | 9 | 9 | 5 | | 10 | 4 | 3 | | 11 | 1 | 1 | | 12 | 0 | 0 |

For since the velocity with which the surface \( AB \) descends, the area of that surface being always the same, is as the square roots of its altitude above the orifice (Prop. I. Cor. 6); and since the velocities are as the times of description, the times will also be as the square roots of the altitudes, that is, when

12, 11, 10, 9, &c. are the times,

144, 121, 100, 81 will be the altitudes of the surface.

Prop. VII.

198. To explain the theory and construction of clepsydra or water-clocks.

A clepsydra, or water-clock, is a machine which, filled with water, measures time by the descent of the fluid surface. See Part III. on Hydraulic Machinery.

It has already been demonstrated in Prop. IV., that the times in which the surface \( AB \) descends to \( DE \) and \( RT \), &c. are as the areas \( GM \) \( rH \), \( GM \) \( rU \), &c. If such a form therefore is given to the vessel that the areas \( GM \) \( rH \), \( GM \) \( rU \), &c. increase uniformly as the times, or are to one another as the numbers 1, 2, 3, 4, 5, &c. the times in which the surface \( AB \) descends to \( DE \), and \( RT \), &c. will be in the same ratio, and the vessel will form a machine for measuring time. If the vessel is cylindrical and empties itself in 12 hours, its altitude may be divided in such a manner that the fluid surface may take exactly an hour to descend through each division. Let the cylindrical vessel, for example, be divided into 144 equal parts, then the surface of the water, when the twelve hours begin to run, will be 144 parts above the bottom of the vessel; when one hour is completed, the surface will be 121 parts above the bottom and so on in the following manner:

Prop. VIII.

199. To explain the lateral communication of motion in fluids.

This property of fluids in motion was discovered by M. Venturi, Professor of Natural Philosophy in the University of Modena, who has illustrated it by a variety of experiments in his work on the lateral communication of motion in fluids. Let a pipe \( AC \), about half an inch in diameter and a foot long, proceeding from the reservoir \( AB \), and having its extremity bent into the form \( CD \), be inserted into the vessel \( CDG \), whose side \( DG \) gradually rises till it passes over the rim of the vessel. Fill this vessel with water, and pour the same fluid into the reservoir \( AB \), till running down the pipe \( AC \), it forms the stream \( EGH \). In a short while, the water in the vessel \( CDG \) will be carried off by the current \( EG \), which communicates its motion to the adjacent fluid. In the same way, when a stream of water runs through air, it drags the air along with it, and produces wind. Hence we have the water blowing machine, which conveys a blast to furnaces, and which will blowing be described in a future part of this article. The lateral communication of motion, whether the surrounding fluid be air or water, is well illustrated by the following beautiful experiments of Venturi's. In the side of the reservoir \( AB \) (fig. 65), insert the horizontal pipe \( P \) about an inch Motion of and a half in diameter, and five inches long. At the point Fluids, &c., of this pipe, about seven-tenths of an inch from the reservoir, fasten the bent glass tube \( o n m \), whose cavity communicates with that of the pipe, whilst its other extremity is immersed in coloured water contained in the small vessel \( F \). When water is poured into the reservoir \( A B \), having no connection with the pipe \( C \), so that it may issue from the horizontal pipe, the red liquor will rise towards \( m \) in the incurvated tube \( o n m \). If the descending leg of this glass syphon be six inches and a half longer than the other, the red liquor will rise to the very top of the syphon, enter the pipe \( P \), and running out with the other water, will in a short time leave the vessel \( F \) empty. Now, the cause of this phenomenon is evidently this: When the water begins to flow from the pipe \( P \), it communicates with the air in the syphon \( o n m \), and drags a portion along with it. The air in the syphon is therefore rarefied, and this process of rarefaction is constantly going on as long as the water runs through the horizontal pipe. The equilibrium between the external air pressing upon the fluid in the vessel \( F \), and that included in the syphon, being thus destroyed, the red liquor will rise in the syphon, till it communicates with the issuing fluid, and is dragged along with it through the orifice of the pipe \( P \), till the vessel \( F \) is emptied.

**Prop. IX.**

200. To find the horizontal distance to which fluids will spout from an orifice perforated in the side of a vessel, and the curve which it will describe.

Let \( A B \) be a vessel filled with water, and \( C \) an orifice in its side, so inclined to the horizon as to discharge the fluid in the direction \( C P \). If the issuing fluid were influenced by no other force except that which impels it out of the orifice, it would move with an uniform motion in the direction \( C P \). But immediately upon its exit from the orifice \( C \), it is subject to the force of gravity, and is therefore influenced by two forces, one of which impels it in the direction \( C P \), and the other draws it downwards in vertical lines. Make \( C E \) equal to \( E G \), and \( C P \) double of \( C S \), the altitude of the fluid. Draw \( P L \) parallel to \( C K \), and join \( S L \). Draw also \( E F \), \( G H \) parallel to \( C N \), and \( F M \), \( H N \) parallel to \( C G \), Motions of and let \( C M \), \( C N \) represent the force of gravity, or the spaces through which it would cause a portion of fluid to descend in the time that this portion would move through \( C E \), \( C G \) respectively, by virtue of the impulsive force. Now, it follows from the composition of forces (Dynamics, 135), that the fluid at \( C \), being solicited in the direction \( C E \) by a force which would carry it through \( C E \) in the same time that the force of gravity would make it fall through \( C M \), will describe the diagonal \( C F \) of the parallelogram \( C E F M \), and will arrive at \( F \) in the same time that it would have reached \( E \) by its impulsive force, or \( M \) by the force of gravity; and for the same reason the portion of the fluid will arrive at \( H \) in the same time that it would have reached \( G \) by the one force, and \( N \) by the other. The fluid therefore being continually deflected from its rectilinear direction \( C P \) by the force of gravity, will describe a curve line \( C F H K \), which will be a parabola; for since the motion along \( C P \) must be uniform, \( C E \), \( C G \) will be to one another as the times in which they are described, and may therefore represent the times in which the fluid would arrive at \( E \) and \( G \), if influenced by no other force. But in the time that the fluid has described \( C E \) gravity has made it fall through \( E F \), and in the time that it would have described \( C G \), gravity has caused it to fall through \( G H \). Now, since the spaces are as the squares of the times in which they are described (Dynamics, 37, 2), we shall have \( E F : G H = C E^2 : C G^2 \). But on account of the parallelograms \( C E F M \), \( C G H N \), \( E F \) and \( G H \) are equal to \( C M \) and \( C N \) respectively, and \( M F \), \( N H \), to \( C E \), \( C G \) respectively; therefore \( C M : C N = M F^2 : N H^2 \), which is the property of the parabola, \( C M \), \( C N \) being the abscisse, and \( M F \), \( N H \) the ordinates (Conic Sections, Part I. Prop. IX. Cor.).

201. On account of the parallels \( L P \), \( C X \), \( L C \), \( G X \), the triangles \( L C P \), \( G C X \) are similar, and therefore (Geom. Sect. IV. Theor. XX.) \( C G : C X = P C : P L \) and \( G X : C X = C L : P L \). Hence \( C G = \frac{C X \times P C}{P L} \), and \( G X = \frac{C X \times C L}{P L} \), but since \( P C = 2 C S \), we have \( C G = \frac{C X \times 2 C S}{P L} \), and since \( G X = G X - H X \), we shall have \( G H = \frac{C X \times C L}{P L} - H X \). But as the parameter of the parabola \( C R K \) is equal to \( 4 C S \) (\(^1\)), we have, by the property of this conic section, \( N H^2 = C N \times 4 C S \), or \( C G^2 = 4 G H \times C S \); therefore, by substituting in this equation the preceding values of \( C G \) and \( G H \), we shall have \( C X^2 \times C S = C X \times C L \times P L - H X \times P L^2 \). Now, it is evident, from this equation, that \( H X \) is nothing, or vanishes when \( C X = 0 \), or when \( C X = \frac{C L \times P L}{C S} \), for \( H X \) being \( = 0 \), \( H X \times P L^2 \), will also be \( = 0 \), and the equation will become \( C X^2 \times C S = C X \times C L \times P L \), or dividing by \( C X \) and \( C S \), it becomes \( C X = \frac{C L \times P L}{C S} \). But,

\(^1\) The parameter of the parabola described by the issuing fluid, is equal to four times the altitude of the fluid above the orifice. For, since the fluid issues at \( C \) with a velocity equal to that acquired by falling through \( S C \), if this velocity were continued uniform, the fluid would move through \( 2 C S \) or \( C P \), in the same time that a heavy body would fall through \( S C \). Draw \( P Q \) parallel to \( C S \), and \( Q W \) to \( C P \); then, since \( Q \) is in the parabola, the fluid will describe \( C P \) uniformly in the same time that it falls through \( C W \) by the force of gravity, therefore \( C W = C S \). Now \( C P = 2 C S \), and \( C P^2 = 4 C S^2 = 4 \times C S \times C S = 4 \times C S \times C W \); but it is a property of the parabola, that the square of the ordinate \( W Q \) or \( C P \) is equal to the product of the abscissa \( C W \) and the parameter, therefore \( 4 C S \) is the parameter of the parabola.

otion of when HX vanishes towards K, CX is equal to CK, consequently CK = \(\frac{CL \times PL}{CS}\). Bisect CK in T, then CT = \(\frac{CK}{2}\), and \(CT = \frac{CL \times PL}{2 \times CS}\). Draw TR perpendicular to CK, and TR will be found = \(\frac{CL^2}{4 \times CS}\). Then, if H m be drawn at right angles to HX, we shall have CX = CT — \(Hm = \frac{CL \times PL}{2 \times CS} - Hm\) and HX = RT — R = \(\frac{CL^2}{4 \times CS} - Rm\). After substituting these values of CX and HX in the equation \(CX^2 \times CS = CX \times CL \times PL - HX \times PL^2\), it will become, after the necessary reductions, \(Hm^2 = \frac{PL^2}{CS} \times Rm\). The curve CRK is therefore a parabola whose vertex is R, its axis RT, and its parameter \(\frac{PL^2}{CS}\), R m being an abscissa of the axis, and H m its corresponding ordinate. Now, making \(a = CS\), the altitude of the reservoir; \(R = radius\); \(m = PL\) the sine of the angle PCL; and \(n = CL\), the cosine of the same angle, CP being radius. Then CP : PL = R : m, therefore \(PL \times R = CP \times m\), and dividing by R and substituting \(2a\) or \(2CS\), instead of its equal CP, we have \(PL = \frac{2a}{R}\), and by the very same reasoning, we have \(CL = \frac{2an}{R}\).

Hence \(RT = \frac{CL^2}{4 \times CS}\) will be \(= \frac{4a^2}{R^2}\) divided by \(4a\), or \(RT = a \times \frac{n^2}{R^2}\) and \(CT = \frac{CL \times PL}{2 \times CS} = \frac{4a^2}{2a \times R^2} = 2a \times \frac{m}{R^2}\), and the parameter of the parabola \(= \frac{PL^2}{CS} = \frac{4a^2}{a \times R^2} = 4a \times \frac{m^2}{R^2}\).

202. Hence we have the following construction. With \(\frac{1}{2}CS\) as radius, describe the semicircle SGC, which the

direction CR of the jet or issuing fluid meets in G. Draw GN perpendicularly to CS, and having prolonged it towards R, make GR equal to GN. From R let fall RT perpendicular to CK, and meeting it in T, and upon RT, CT, describe the parabola CRK having its vertex in R, this parabola will be the course of the issuing fluid. For, by the construction NR or CT = 2 GN, and on account of the similar triangles SGC, CGN, SC : SG = CG : GN; hence \(SC \times GN = SG \times CG\), or \(2GN\), or \(CT = \frac{2SG \times CG}{SC}\).

But from the similarity of triangles CS : CG = SG : GN and CS : CG = CG : CN, consequently, when CG is radius or = R, GN will be the sine \(m\) of the angle GCS, and CN its cosine \(n\); and we shall then have, by Euclid VI. 16.

and reduction \(SG = \frac{CS \times m}{R}\), and \(CG = \frac{CS \times n}{R}\). By substituting these values of SG and CG in the equation \(CT = \frac{2SG \times CG}{SC}\), we have \(CT = \frac{2}{SC} \times \frac{CS \times m}{R} \times \frac{CS \times n}{R} = \frac{2CS \times mn}{R^2} = 2a \times \frac{mn}{R^2}\). But the parameter P of the parabola CRK is equal to \(\frac{CT^2}{RT}\), because it is a third proportional to the abscissa and its ordinate, therefore \(P = \frac{4a^2 \times m^2}{R^2 \times RT}\). Now \(RT = CN\), and \(CN = \frac{NG \times n}{m}\), because \(CN : NG = m : n\), or \(CN = RT = a \times \frac{n^2}{R^2}\) by substituting the preceding value of NG. Therefore, the parameter \(P = \left(\frac{4a^2 \times m^2}{R^4}\right)\div \left(\frac{a \times n^2}{R^2}\right) = 4a \times \frac{m^2}{R^2}\), which is the same value of the parameter as was found in the preceding article, and therefore verifies the construction.

203. Cor. 1. Since NG = GR and CT = TK, the amplitude or distance CT, to which the fluid will reach on a horizontal plane, will be 4 NG, or quadruple the sine of the angle formed by the direction of the jet and a vertical line, the chord of the arch CG, being radius.

204. Cor. 2. If S n be made equal to CN, and n g be drawn parallel to CT, and g r be made equal to n g; then, if the direction of the jet be C g, the fluid will describe the parabola C r K whose vertex is r, and will meet the horizontal line in K, because n g = NG, and 4 n g = 4 NG = CK. The same may be shown of every other pair of parabolas, whose vertices R r are equidistant from a c, a horizontal line passing through the centre of the circle.

205. Cor. 3. Draw the ordinate a b through the centre a, and since this is the greatest ordinate that can be drawn, the distance to which the water will spout, being equal to \(4a\), will be the greatest when its line of direction passes through b, that is, when it makes an angle of 45° with the horizon.

206. Cor. 4. If an orifice be made in the vessel AB at N, and the water issues horizontally in the direction NG, it will describe the parabola NT, and CT will be equal to 2 NG. For (by Prop. IX., note) the parameter of the parabola NT is equal to 4 NS, and by the property of the parabola \(CT^2 = NC \times 4NS\), or \(CT = 2 \sqrt{NC \times NS}\); but by the property of the circle (Geom. Sect. IV. Theor. XXVIII.) \(NG^2 = NC \times NS\), and \(NG = \sqrt{NC \times NS}\), hence \(CT = 2 NG\). If the fluid is discharged from the orifice at n, so that \(S n = CN\), n g will be = NG, and it will spout to the same distance CT.

Prop. X.

207. To determine the pressure exerted upon pipes by the water which flows through them.

Let us suppose the column of fluid CD divided into an infinite number of laminae EF f c. Then friction being abstracted, every particle of each lamina will move with the same velocity when the pipe CD is horizontal. Now, the velocity at the vena contracta \(m\) may be expressed by \(\sqrt{A}\), A being the altitude of the fluid in the reservoir. But the velocity at the vena contracta is to the velocity in the pipe, as the area of the latter is to the area of the former. Therefore \( \delta \) being the diameter of the vena contracta, and \( d \) that of the pipe CD, the area of the one will be to the area of the other, as \( \delta^2 : d^2 \). (Geometry, Sect. VI. Prop. IV.) consequently we shall have \( d^2 : \delta^2 = \sqrt{A} : \frac{\delta^2}{d^2} \), the velocity of the water in the pipe. But since the velocity \( \sqrt{A} \) is due to the altitude \( A \), the velocity \( \frac{\delta^2}{d^2} \sqrt{A} \) will be due to the altitude \( \frac{\delta^4}{d^4} \). Now, as each particle of fluid which successively reaches the extremity DH of the pipe, has a tendency to move with the velocity \( \sqrt{A} \), while it moves only with the velocity \( \frac{\delta^2}{d^2} \sqrt{A} \), the extremity D of the pipe will sustain a pressure equal to the difference of the pressures produced by the velocities \( \sqrt{A} \) and \( \frac{\delta^2}{d^2} \sqrt{A} \), that is, by a pressure \( A - \frac{\delta^4}{d^4} \), \( A \) representing the pressure which produces the velocity \( \sqrt{A} \), and \( \frac{\delta^4}{d^4} \) the pressure which produces the velocity \( \frac{\delta^2}{d^2} \sqrt{A} \).

But this pressure is distributed through every part of the pipe CD; consequently the pressure sustained by the sides of the pipe will be \( A - \frac{\delta^4}{d^4} \).

208. Cor. 1. If a very small aperture be made in the side of the pipe, the water will issue with a velocity due to the height \( A - \frac{\delta^4}{d^4} \). When the diameter \( \delta \) of the orifice is equal to the diameter \( d \) of the pipe, the altitude becomes \( A - A \) or nothing; and if the orifice is in this case below the pipe, the water will descend through it by drops. Hence we see the mistake of those who have maintained, that when a lateral orifice is pierced in the side of a pipe, the water will rise to a height due to the velocity of the included water.

209. Cor. 2. Since the quantities of water, discharged by the same orifice, are proportional to the square roots of the altitudes of the reservoir, or to the pressures exerted at the orifice, the quantity of water discharged by a lateral orifice may be easily found. Let \( W \) be the quantity of water discharged in a given time by the proposed aperture under the pressure \( A \), and let \( w \) be the quantity discharged under the pressure \( A - \frac{\delta^4}{d^4} \). Then \( W : w = \sqrt{A} : \sqrt{A - \frac{\delta^4}{d^4}} \), consequently, \( w \times \sqrt{A} = W \times \sqrt{A - \frac{\delta^4}{d^4}} \) and \( w = \frac{W \times \sqrt{A - \frac{\delta^4}{d^4}}}{\sqrt{A}} = W \sqrt{\frac{d^4 - \delta^4}{d^2}} \). Therefore, since \( W \) may be determined by the experiments in the following chapter, \( w \) is known.

CHAPTER II.—ACCOUNT OF EXPERIMENTS FROM VESSELS, EITHER BY ORIFICES OR ADDITIONAL TUBES, OR RUNNING IN PIPES OR OPEN CANALS.

210. In the preceding chapter, we have taken notice of the contraction produced upon the vein of fluid issuing from an orifice in a thin plate, and have endeavoured to ascertain its cause. According to Sir Isaac Newton, the diameter of the orifice of the vena contracta is to that of the orifice as 21 to 25. Poleni makes it as 11 to 13; Bernoulli as 5 to 7; the Chevalier de Buat as 6 to 9; Bossut as 41 to 50; Michelotti as 4 to 5; Venturi as 4 to 5; Bidone as 33 to 50; and Eydeltein as 32 to 50. This ratio, however, is by no means constant. It varies with the form and position of the orifice, with the thickness of the plate in which the orifice is made, and likewise with the form of the vessel and the weight of the superincumbent fluid. But these variations are too trifling to be regarded in practice.—We shall now lay before the reader an account of the results of the experiments of different philosophers, but particularly those of the Abbé Bossut, to whom the science is deeply indebted both for the accuracy and extent of his labours.

Sect. I. On the Quantity of Water discharged from Vessels constantly full by Orifices in thin Plates.

211. The following table contains the results obtained by Michelotti when the orifices are vertical, and of a square or circular form, the altitude of the head of water varying from 6 feet to about 22 feet.

| Altitude of Water above the centre of the orifice | Size and form of the orifice | Time of running | Cubic feet of water discharged | Michelotti's experiments | |--------------------------------------------------|-----------------------------|----------------|-------------------------------|------------------------| | Ft. In. Lin. Pts. | | | | | | 6 7 4 3 | Square | 10 | 463 7 3 | | | 11 8 1 6 | Square | 12 | 566 5 6 | | | 11 9 9 10 | Square | 10 | 612 1 5 | | | 21 8 3 6 | Square | 5 | 415 5 3 | | | 21 8 7 0 | Square | 6 | 499 2 8 | | | 6 7 6 0 | Square | 15 | 329 9 8 | | | 11 5 1 4 | Square | 15 | 423 5 7 | | | 21 5 3 7 | Square | 10 | 385 4 0 | | | 6 9 1 0 | Square | 30 | 158 6 7 | | | 11 10 8 1 | Square | 24 | 163 9 6 | | | 21 6 1 0 | Square | 60 | 562 11 4 | | | 6 8 4 0 | Circular of | 15 | 542 10 6 | | | 11 7 1 0 | Circular of | 12 | 570 11 8 | | | 21 7 4 0 | Circular of | 8 | 521 3 7 | | | 6 9 5 0 | Circular of | 30 | 488 8 3 | | | 11 8 8 0 | Circular of | 28 | 589 6 5 | | | 21 10 10 0 | Circular of | 20 | 575 5 10 | | | 6 10 6 0 | Circular of | 60 | 247 4 3 | | | 11 8 11 0 | Circular of | 60 | 324 1 5 | | | 22 0 2 0 | Circular of | 60 | 444 6 5 | | The coefficient obtained from these experiments is 0.625.

212. Messrs Brindley and Smeaton found that 20 cubic feet of water were discharged from orifices 1 inch square, in the following times, varying with the height of the water.

| Height of water in feet | Time of discharging 20 cubic feet | |------------------------|----------------------------------| | 1 | 562 seconds | | 2 | 400 | | 3 | 320 | | 4 | 284 | | 5 | 254 |

When the height of the water was 6 feet, and the orifice an inch square, 20 cubic feet were discharged in 17 minutes 33 seconds. The coefficient obtained from these experiments is 0.63.

213. When the water stands always at the upper surface of a rectangular aperture, without the upper edge, the aperture is called a notch. The following table shows the time of discharging 20 cubic feet, through notches 6 inches wide, of various depths.

| Depth of the notch in inches | Time of discharging 20 cubic feet | |-----------------------------|----------------------------------| | 1 | 436 seconds | | 1½ | 293 | | 2½ | 139 | | 3½ | 93 | | 6½ | 30 | | 5 | 46 | | 1½ | 326 | | 5½ | 230 | | 5½ | 47 |

214. In the following experiments by the Abbé Bossut which were frequently repeated in various ways, the orifice was pierced in a plate of copper about half a line thick. When the orifice is in the bottom of the vessel, it is called a horizontal orifice; and when it is in the side of it, it is called a lateral orifice.

Table II. Shewing the Quantity of Water discharged in one minute, by orifices differing in form and position.

| Altitude of the fluid above the centre of the orifice | Form and position of the orifice | The orifice's diameter | No. of cub. in discharged in a minute | |------------------------------------------------------|---------------------------------|------------------------|-------------------------------------| | Ft. In. Lin. | | | | | 11 8 10 | Circular and Horizontal | 6 lines | 2311 | | | Circular and Horizontal | 1 inch | 9281 | | | Circular and Horizontal | 2 inches | 37203 | | | Rectangular and Horizontal | 1 inch by 3 lines | 2933 | | | Horizontal and Square | 1 inch, side | 11817 | | | Horizontal and Square | 2 inch, side | 47361 | | 9 0 0 | Lateral and Circular | 6 lines | 2018 | | | Lateral and Circular | 1 inch | 8135 | | | Lateral and Circular | 6 lines | 1353 | | | Lateral and Circular | 1 inch | 5436 | | | Lateral and Circular | 1 inch | 628 |

215. From the results contained in the preceding table, we may draw the following conclusions:

1. That the quantities of water discharged in equal times by different apertures, the altitudes of the fluid being the same, are very nearly as the areas of the orifices. That is, if $A$ or $a$ represent the areas of the orifices, and $W$, $w$ the quantities of water discharged,

$$ W : w = A : a $$

2. The quantities discharged in equal times by the same aperture, the altitude of the fluid being different, are to one another very nearly as the square roots of the altitudes of the water in the reservoir, reckoning from the centres of the orifices. That is, if $H$, $h$ be the different altitudes of the fluid, we shall have

$$ W : w = \sqrt{H} : \sqrt{h} $$

3. Hence we may conclude in general, that the quantities discharged in the same time by different apertures, and under different altitudes in the reservoir, are in the compound ratio of the areas of the orifices, and the square roots of the altitudes. Thus, if $W$, $w$ be the quantities discharged in the same time from the orifices $A$, $a$, under the same altitude of water; and if $W'$, $w'$ be the quantities discharged in the same time by the same aperture $a$ under different altitudes, $H$, $h$; then by the first of the two preceding articles

$$ W : w = A : a $$

and by the second

$$ w : W' = \sqrt{H} : \sqrt{h} $$

Multiplying these analogies together, gives us

$$ Ww : W'w = A\sqrt{H} : a\sqrt{h} $$

and by dividing by $w$,

$$ W : W' = A\sqrt{H} : a\sqrt{h} $$

This rule is sufficiently correct in practice; but when great accuracy is required, the following remarks must be attended to.

4. Small orifices discharge less water in proportion than great ones, the altitude of the fluid being the same. The circumferences of the small orifices being greater in proportion to the issuing column of fluid than the circumferences of greater ones, the friction, which increases with the area of the rubbing surfaces, will also be greater, and will therefore diminish the velocity, and consequently the quantity discharged.

5. Hence of several orifices whose areas are equal, that which has the smallest circumference will discharge more water than the rest under the same altitude of fluid in the reservoir, because in this case the friction will be least. Circular orifices, therefore, are the most advantageous of all, for the circumference of a circle is the shortest of all lines that can be employed to inclose a given space.

6. In consequence of a small increase which the contraction of the vein of fluid undergoes, in proportion as the altitude of the water in the reservoir augment, the quantity discharged ought also to diminish a little as that altitude increases.

By attending to the preceding observations, the results of theory may be so corrected, that the quantities of water discharged in a given time may be determined with the greatest accuracy possible.

216. The Abbé Bossut has given the following table containing a comparison of the theoretical with the real discharges, for an orifice one inch in diameter, and for different altitudes of the fluid in the reservoir. The real discharges were not found immediately by experiment, but were determined by the precautions pointed out in the preceding articles, and may be regarded to be as accurate from circular orifices as if direct experiments had been employed. The fourth column was computed by M. Prony. TABLE III. Comparison of the Theoretic with the Real Discharges from an Orifice one inch in diameter.

| Paris Feet | Cubic Inches | Cubic Inches | Ratio of the theoretical to the real discharges | |------------|-------------|-------------|-----------------------------------------------| | 1 | 4381 | 2722 | 1 to 0.62133 | | 2 | 6196 | 3846 | 1 to 0.62073 | | 3 | 7589 | 4710 | 1 to 0.62064 | | 4 | 8763 | 5436 | 1 to 0.62034 | | 5 | 9797 | 6075 | 1 to 0.62010 | | 6 | 10732 | 6654 | 1 to 0.62000 | | 7 | 11592 | 7183 | 1 to 0.61965 | | 8 | 12392 | 7672 | 1 to 0.61911 | | 9 | 13144 | 8135 | 1 to 0.61892 | | 10 | 13855 | 8574 | 1 to 0.61883 | | 11 | 14530 | 8990 | 1 to 0.61873 | | 12 | 15180 | 9384 | 1 to 0.61819 | | 13 | 15797 | 9764 | 1 to 0.61810 | | 14 | 16393 | 10150 | 1 to 0.61795 | | 15 | 16968 | 10472 | 1 to 0.61716 |

Deduction from the preceding table.

217. It is evident from the preceding table, that the theoretical, as well as the real discharges, are nearly proportional to the square roots of the altitudes of the fluid in the reservoir. Thus, if we take the altitudes 1 and 4, whose square roots are as 1 to 2, the real discharges taken from the table are 2722, 5436, which are to one another very nearly as 1 to 2, their real ratio being as 1 to 1.997.

The fourth column of the preceding table also shows us that the theoretical are to the real discharges nearly in the ratio of 1 to 0.62, or more accurately, as 1 to 0.61958; therefore 0.62 is the number by which we must multiply the discharges as found by the formulae in the preceding chapter, in order to have the quantities of water actually discharged.

218. In order to find the quantities of fluid discharged by orifices of different sizes, and under different altitudes of water in the reservoir, we must use the table in the following manner. Let it be required, for example, to find the quantity of water furnished by an orifice three inches in diameter, the altitude of the water in the reservoir being 30 feet. As the real discharges are in the compound ratio of the area of the orifices, and the square roots of the altitudes of the fluid (art. 215, No. 3), and as the theoretical quantity of water discharged by an orifice one inch in diameter, is by the second column of the table 16918 cubic inches in a minute, we shall have this analogy, \( \frac{1}{\sqrt{3}} : \frac{9}{\sqrt{30}} = 16918 : 215961 \) cubic inches, the quantity required. This quantity being diminished in the ratio of 1 to .62, being the ratio of the theoretical to the actual discharges, gives 133896 for the real quantity of water discharged by the given orifice. But (by No. 5 of art. 215), the quantity discharged ought to be a little greater than 133896, because greater orifices discharge more than small ones; and by No. 6 the quantity ought to be less than 133896, because the altitude of the fluid is double that in the table. These two causes therefore having a tendency to increase and diminish the quantity deduced from the preceding table, we may regard 133896 as very near the truth. Had the orifice been less than one inch, or the altitude less than 15 feet, it would have been necessary to diminish the preceding answer by a few cubic inches. Since the velocities of the issuing fluid are as the quantities discharged, the preceding results may be employed also to find the real velocities from those which are deduced from theory.

219. As the velocity of falling bodies is 16.087 feet per second, the velocity due to 16.087 feet will be 32.174 times the feet per second, and as the velocities are as the square velocity, roots of the height, we shall have \( \sqrt{16.087} : \sqrt{H} = 32.174 : V \) the velocity due to any other height, consequently \( V = \frac{32.174 \sqrt{H}}{\sqrt{16.087}} = \frac{32.174 \sqrt{H}}{4.011} = 8.016 \sqrt{H} \), so that 8.016 is the coefficient by which we must always multiply the altitude of the fluid in order to have its theoretical velocity.

220. The following are the coefficients according to various authors, or the ratio of the theoretical to the real discharges from a circular orifice:

- Michelotti ........................................... 0.649 - Borda .................................................. 0.625 - Venturi .................................................. 0.646 - Eytelwein ................................................ 0.640 - Hachette .................................................. 0.690 - Newton .................................................... 0.707 - Helsham .................................................. 0.705 - Brindley .................................................. 0.631 - Smeaton .................................................. 0.631 - Banks ...................................................... 0.750 - Bossut ..................................................... 0.610 - Rennie¹ .................................................... 0.621

Sect. II. On the Quantity of Water discharged from Vessels constantly full, by small Tubes adapted to Circular Orifices.

221. The difference between the actual discharges, and quantities those deduced from theory, arises from the contraction of the fluid vein, and from the friction of the water against the circumference of the orifice. If the operation of any tubes of these causes could be prevented, the quantities of water actually discharged would approach nearer the theoretical discharges. There is no probability of diminishing friction in the present case by the application of unguents; but if a short cylindrical tube be inserted in the orifice of the vessel, the water will follow the sides of the tube, the contraction of the fluid vein will be in a great measure prevented, and the actual discharges will approximate much nearer to those deduced from theory, than when the fluid issues through a simple orifice.

222. If a cylindrical tube two inches long, and two inches in diameter, be inserted in the reservoir, and if this cylindrical orifice is stopped by a piston till the reservoir is filled with water, the fluid, when permitted to escape, will not follow the sides of the tube, that is, the tube will not be filled diameter with water, and the contraction in the vein of fluid will then take place in the same manner as if the orifice were pierced vein in a thin plate. When the cylindrical tube was one inch diameter, and two inches long, the water followed the simple orifices sides of the tube, and the vein of fluid ceased to contract.

While M. Bossut was repeating this experiment, he prevented the escape of the fluid by placing the instrument

¹ Philosophical Transactions, 1631. 223. Table IV. Showing the Quantities of Water discharged by Cylindrical Tubes one inch in diameter with different lengths.

| Variable lengths of the tubes expressed in lines. | Cubic inches discharged in a minute. | |--------------------------------------------------|-------------------------------------| | The tube being filled with the issuing fluid. | 12274 | | The tube not filled with the issuing fluid. | 9282 |

The experiments in the preceding table were made with tubes inserted in the bottom of the vessel. When the tubes were fixed horizontally in the side of the reservoir, they furnished the very same quantities of fluid, their dimensions and the altitude of the fluid remaining the same.

It appears from the preceding results, that the quantities of water discharged increase with the length of the tube, and that these quantities are very nearly as the square roots of the altitudes of the fluid above the interior orifice of the vertical tube.

We have already seen that the theoretical are to the real discharges, as 1 to 0.62, or nearly as 16.1 to 10. But by comparing the two last experiments in the preceding table, it appears that the quantity of fluid discharged by a cylindrical tube where the water follows its sides, is to the quantity discharged by the same tube when the vena contracta is formed, as 13 to 10; and since the same quantity must be discharged by the latter method as by a simple orifice, we may conclude that the quantity discharged according to theory, and that which is discharged by a cylindrical tube and by a simple orifice, are to one another very nearly as the numbers 16, 13, 10. Though the water therefore follows the sides of the cylindrical tube, the contraction of the fluid vein is not wholly destroyed; for the difference between the quantity discharged in this case, and that deduced from theory, is too great to be ascribed to the increase of friction which arises from the water following the circumference of the tube.

224. In order to determine the effect of tubes of different diameters, under different altitudes of water in the reservoir, M. Bossut instituted the experiments, the results of which are exhibited in the following table.

Table V. Showing the Quantities of Water discharged by Cylindrical Tubes two inches long, with different Diameters.

| Constant altitude of the water above the orifice. | Diameter of the Tube. | Quantity of water discharged in a minute. | |--------------------------------------------------|----------------------|------------------------------------------| | Feet. Inches. | Lines. | Cubic inches. | | 3 10 | | | | The tube being filled with the issuing fluid. | 6 | 1689 | | The tube not filled with the issuing fluid. | 6 | 4703 | | 2 0 | | | | The tube being filled with the issuing fluid. | 6 | 1293 | | The tube not filled with the issuing fluid. | 6 | 3598 | | | | | | | | |

225. By comparing the different numbers in this table, we may conclude,

1. That the quantities of water discharged by different cylindrical tubes of the same length, the altitude of the fluid remaining the same, are nearly as the areas of the orifices, or the squares of their diameters.

2. That the quantities discharged by cylindrical tubes of the same diameter and length, are nearly as the square roots of the altitude of the fluid in the reservoir.

3. Hence the quantities discharged during the same time, by tubes of different diameters, under different altitudes of fluid in the reservoir, are nearly in the compound ratio of the squares of the diameters of the tube, and the square roots of the altitudes of the water in the reservoir.

4. By comparing these results with those which were deduced from the experiments with simple orifices, it will be seen that the discharges follow the same laws in cylindrical tubes as in simple orifices.

226. Table VI. Comparison of the Theoretical with the Real Discharges from a Cylindrical Tube one inch in Diameter, and two inches long.

| Constant altitude of the water in the reservoir above the centre of the orifice. | Theoretical discharges through a circular orifice one inch in diameter. | Real discharges in the same time by a cylindrical tube one inch in diameter and two inches long. | Ratio of the theoretical to the real discharges. | |-------------------------------------------------------------------------------|-----------------------------------------------------------------|-------------------------------------------------|-----------------------------------------------| | Paris Feet. | Cubic Inches. | Cubic Inches. | | | 1 | 4381 | 3539 | 1 to 0.81781 | | 2 | 6196 | 5002 | 1 to 0.80729 | | 3 | 7589 | 6126 | 1 to 0.80724 | | 4 | 8763 | 7070 | 1 to 0.80681 | | 5 | 9797 | 7900 | 1 to 0.80638 | | 6 | 10732 | 8534 | 1 to 0.80633 | | 7 | 11592 | 9340 | 1 to 0.80573 | | 8 | 12392 | 9975 | 1 to 0.80496 | | 9 | 13144 | 10579 | 1 to 0.80485 | | 10 | 13855 | 11151 | 1 to 0.80483 | | 11 | 14530 | 11693 | 1 to 0.80477 | | 12 | 15180 | 12205 | 1 to 0.80403 | | 13 | 15797 | 12609 | 1 to 0.80390 | | 14 | 16393 | 13177 | 1 to 0.80382 | | 15 | 16968 | 13620 | 1 to 0.80270 |

Comparison of the theoretical with the real discharges in cylindrical tubes. The above table is deduced from the foregoing experiments, and contains a comparative view of the quantities of water discharged by a simple orifice, according to theory, and those discharged by a cylindrical tube of the same diameter under different altitudes of water. The numbers might have been more accurate by attending to some of the preceding remarks; but they are sufficiently exact for any practical purpose. The fourth column, containing the ratio between the theoretical and actual discharges, was computed by M. Prony.

By comparing the preceding table with that in art. 216, we shall find that cylindrical tubes discharge a much greater quantity of water than simple orifices of the same diameter, and that the quantities discharged are as 81 to 62 nearly. This is a curious phenomenon, and will be afterwards explained.

227. The application of this table to other additional tubes under different altitudes of the fluid, not contained in the first column, is very simple. Let it be required, for example, to find the quantity of water discharged by a cylindrical tube, 4 inches in diameter, and 8 inches long, the altitude of the fluid in the reservoir being 25 feet. In order to resolve this question, find (by art. 218) the theoretical quantity discharged, which in the present instance will be 350490 cubic inches, and this number diminished in the ratio of 1 to 0.81 will give 284773 for the quantity required. The length of the tube in this example was made 8 inches, because, when the length of the tube is less than twice its diameter, the water does not easily follow its interior circumference. If the tube were longer than 8 inches, the quantity of fluid discharged would have been greater, because it uniformly increases with the length of the tube; the greatest length of the tube being always small, in comparison with the altitude of the fluid in the reservoir.

228. Hitherto we have supposed the tube to be exactly cylindrical. When its interior surface, however, is conical, the quantities discharged undergo a considerable variation, which may be estimated from the following experiments of the Marquis Poleni, published in his work, De Castellis per quae dericeantur Fluviorum aqua, which appeared at Padua in 1718.

| Apertures employed | Interior diameter | Exterior diameter | Quantity discharged in a min. in cubic feet | Time in which 73935 cubs. in. were discharged | |--------------------|------------------|-------------------|------------------------------------------|---------------------------------------------| | Constant altitude of the water in the reservoir, 256 lines, or 1 foot 9 inches and 4 lines. | Orifice in a thin plate, Cylindrical tube, 1st Conical tube, 2nd Conical tube, 3rd Conical tube, 4th Conical tube | 26 lines | 26 lines | 15877 | 4' 36" | | Length of each tube | 92 lines, or 7 inches and 8 lines. | 26 | 26 | 23434 | 3' 7" | | | | 33 | 26 | 24758 | 2' 57" | | | | 42 | 26 | 24619 | 2' 58" | | | | 60 | 26 | 24345 | 3' 0" | | | | 118 | 26 | 23687 | 3' 5" |

From these experiments we are authorized to conclude, 1. That the real discharges are less than those deduced from theory, which in the present case is 27425 cubic inches in a minute; and, 2. That when the interior orifice of the tube is enlarged to a certain degree, the quantity discharged is increased; but that when this enlargement is too great, a contraction takes place without the exterior orifice, and the quantity discharged suffers a diminution. If the smallest base of the conical tube be inserted in the side of the reservoir, it will furnish more water than a cylindrical tube whose diameter is equal to the smallest diameter of the conical tube; for the divergency of its sides changes the oblique motion which the particles would otherwise have had, when passing from the reservoir into the tube.

229. The experiments of Poleni and Bossut having been made only with tubes of a conical and cylindrical form, M. Venturi was induced to institute a set of experiments, in which he employed tubes of the various forms exhibited in fig. 69. The results of his researches are contained in the following table, for which we have computed the column containing the number of cubic inches discharged in one minute, in order that the experiments of the Italian philosopher may be more easily compared with those which are exhibited in the preceding tables. The constant altitude of the water in the reservoir was 32.5 French inches or 34.642 English inches. The quantity of water which flowed out of the vessel in the times contained in the first column was 4 French cubic feet, or 4.845 English cubic feet. The measures in the table are all English, unless the contrary be expressed. **TABLE VIII. Shewing the Quantities of Water discharged from Orifices of various forms, the constant Altitude of the Fluid being 32.5 French, or 34.642 English inches.**

| No. | Nature and dimensions of the tubes and orifices. | Time in which 4 Paris cubic feet were discharged. | Paris cubic inches discharged in a minute. | |-----|-------------------------------------------------|--------------------------------------------------|------------------------------------------| | 1 | A simple circular orifice in a thin plate, the diameter of the aperture being 1.6 inches. | Seconds. 41 | 10115 | | 2 | A cylindrical tube 1.6 inches in diameter, and 4.8 inches long. | Seconds. 31 | 13378 | | 3 | A tube similar to B, fig. 69, which differs from the preceding only in having the contraction in the shape of the natural contracted vein. | Seconds. 31 | 13378 | | 4 | The short conical adjujate A, fig. 69, being the first conical part of the preceding tube. | Seconds. 42 | 9874 | | 5 | The tube D, fig. 69, being a cylindrical tube adapted to the small conical end A, m n being 3.2 inches long. | Seconds. 42.5 | 9758 | | 6 | The same adjujate, m n being 12.8 inches. | Seconds. 45 | 9216 | | 7 | The same adjujate, m n being 23.6 inches. | Seconds. 48 | 8640 | | 8 | The tube C, consisting of the cylindrical tube of Exp. 2, placed over the conical part of A. | Seconds. 32.5 | 12760 | | 9 | The double conical pipe E, a b = a c = 1.6 inches, e d = 0.977 inches, e f = 1.376 inches, and the length e e of the outer cone = 4.351 inches. | Seconds. 27.5 | 15081 | | 10 | The tube F, consisting of a cylindrical tube 3.2 inches long, and 1.376 inches in diameter, interposed between the two conical parts of the preceding. | Seconds. 28.5 | 14516 |

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230. These experiments of Venturi inform us of a curious deductive fact, extremely useful to the practical hydraulist. They incontestably prove, that when water is conveyed through a straight cylindrical pipe of an unlimited length, the discharge of water may be increased only by altering the form of the terminations of the pipe; that is, by making the end of the pipe A, fig. 70, of the same form as the *vena contracta*, the ratio of the discharges was as 0.92 to 1.00, and when its edges were rounded off, as 0.98 to 1.00 computing from its least section. He found also that the smallest quantity of water was discharged, when the interior extremity of the tube projected within the reservoir, the quantity furnished in this case being reduced to one half of what was discharged when the tube had its proper position.

233. When a cylindrical tube is applied to an orifice, Reason the oblique motion of the particles which enter it is diminished; the vertical velocity of the particles, therefore, is diminished, and consequently the quantity of water discharged. M. Venturi maintains that the pressure of the atmosphere thus reduces the expense of water through a simple cylindrical flow of the tube, and that in conical tubes, the pressure of the atmosphere increases the expenditure in the ratio of the exterior section of the tube to the section of the contracted vein, whatever be the position of the tube.

234. Of all the tubes that can be employed for discharging water, that is the most advantageous which has the form of a contracted vein. Hence, it will be a truncated cone with its greatest base next the reservoir, having its length equal to half the diameter of that base, and the area of the two orifices as 8 to 5, or their diameters in the sub-duplicate ratio of these numbers, viz. as $\sqrt{8} : \sqrt{5}$.

**Sect. III.—Experiments on the Exhaustion of Vessels.**

235. It is almost impossible to determine the exact time in which any vessel of water is completely exhausted. When the surface of the fluid has descended within a few inches of the orifice, a kind of conoidal funnel is formed immediately above the orifice. The pressure of the superincumbent column being therefore removed, the time of exhaustion is prolonged. The water falls in drops; and it is next to impossible to determine the moment when the vessel is empty. Instead, therefore, of endeavouring to ascertain the time in which vessels are completely exhausted, the Abbé Bossuet has determined the times in which the superior surface of the fluid descends through a certain vertical height, and his results will be found in the following table:— TABLE IX. Showing the Times in which Vessels are partly exhausted.

| Primitive altitude of the water in the vessel | Constant area of a horizontal section of the vessel | Diameter of the circular orifice | Depression of the upper surface of the fluid | Time in which this depression takes place | |---------------------------------------------|---------------------------------------------------|---------------------------------|-----------------------------------------------|------------------------------------------| | Paris Feet | Square Feet | Inches | Feet | Min. Sec. | | 11.6666 | 9 | 1 | 4 | 7 | | | | 2 | 4 | 1 | | | | 1 | 9 | 20 | | | | 2 | 9 | 5 |

PG DE √A/7854 PG—PU T

236. In order to compare these experimental results with those deduced from theory, we must employ the formula (in Prop. V. 195) where the time in which the surface descends through any height is \( T = \frac{DE \times \sqrt{PG - PU}}{A \sqrt{16.087}} \), in which DE is the area of a section of the vessel, PG the primitive altitude of the surface above the centre of the orifice, PU the altitude of the surface after the time T is elapsed, A the area of the orifice, and 16.087 the space through which a heavy body descends in one second of time. That the preceding formula may be corrected, we must substitute 0.62 A, or \( \frac{5A}{8} \), instead of A, in the formula, 0.62, A being the area of the vena contracta; and as the measures in the preceding table are in Paris feet, we must use 15.085, instead of 16.087, the former being the distance in Paris feet, and the latter the distance in English feet, which falling bodies describe in a second. The formula, therefore, will become \( T = \frac{DE \times \sqrt{PG - PU}}{0.62 A \sqrt{15.085}} \).

It appears from this table that the times of discharge, by experiment, differ very little from those deduced from the corrected formula; and that the latter always err in defect. This may arise from 0.62 being too great a multiplier for finding the corrected diameter of the orifice. When the orifices are in the sides of the reservoir, the altitude PG, PU of the surface may be reckoned from the centre of gravity of the orifice, unless when it is very large.

Sect. IV. Experiments on Vertical and Oblique Jets.

237. We have already seen that, according to theory, vertical jets should rise to the same altitude as that of the jets do reservoirs from which they are supplied. It will appear, however, from the following experiments of Bossut, that some jets do not rise exactly to this height. This arises from the friction at the orifice, the resistance of the air, and several other causes which shall afterwards be explained.

TABLE XI. Containing the Altitudes to which Jets rise through Adjutages of different forms, the Altitude of the Reservoir being Eleven Feet, reckoning from the upper surface of the horizontal Tubes in P, o p R.

| Diameter of the horizontal tubes m P, o R, each being six feet long | Form of the orifices | References to Fig. 71 | Diameter of the orifice | Altitude of the jet when rising vertically, reckoning from m. | Altitude of the jet when inclined a little to the vertical. | Description of the jets | |---------------------------------------------------------------------|---------------------|----------------------|------------------------|---------------------------------------------------------------|---------------------------------------------------------------|-------------------------| | Inch. Lines | Simple orifice | II | 2 | 10 0 10 | 10 4 6 | The vertical jet beautiful. | | 3 8 | | G | 4 | 10 5 10 | 10 7 6 | The vertical jet beautiful, not much enlarged at the top. | | 3 8 | | F | 8 | 10 6 6 | 10 8 0 | All the jets occasionally rise to different heights. This very perceptible in the present experiment. | | 3 8 | Conical tube | E | 34 by 70 | 9 6 4 | 9 8 6 | The vertical jet much enlarged at top. | | 3 8 | Cylindrical tube | D | 4 by 70 | 9 1 6 | 7 3 6 | The inclined one less so, and more beautiful. | | 0 9½ | Simple orifice | M | 2 | 9 11 0 | | The jet beautiful. | | 0 9½ | L | 4 | 9 7 10 | | The jet much deformed, and very much enlarged at top. | | 0 9½ | K | 8 | 7 10 0 | | The column much broken; and the successive jets detached from each other. | 238. It appears, from the three first experiments of the preceding table, that great jets rise higher than small ones; Fig. 71.

and from the three last experiments, that small jets rise higher than great ones when the horizontal tube is very narrow. There is, therefore, a certain proportion between the diameter of the horizontal tube and that of the adjutage or orifice, which will give a maximum height to the jet. This proportion may be found in the following manner. Let D be the diameter of the tube, d that of the adjutage, a the altitude B m of the reservoir, b the velocity along the tube; and as the velocity at the adjutage is constant, it may be expressed by \( \sqrt{a} \). Now (art. 189, note) the velocity in the tube is to the velocity at the adjutage as the area of their respective sections, that is, as the square of the diameter of the one is to the square of the diameter of the other. Therefore, \( \frac{\sqrt{a}}{\sqrt{b}} = \frac{D^2}{d^2} \), and consequently, \( b = \frac{d^2 \sqrt{a}}{D^2} \). If there is another tube and another adjutage, the corresponding quantities may be the same letters in the Greek character, viz. Δ, δ, α, β, and we shall have the equation \( \beta = \frac{\Delta^2 \sqrt{\alpha}}{\delta^2} \). If we wish, therefore, that the two jets be furnished in the same manner, then if the velocity in the first tube leaves to the first jet all the height possible, the velocity in the second tube leaves also to the second jet all the height possible, and we shall have \( b = \beta \), or \( \frac{d^2 \sqrt{a}}{D^2} = \frac{\Delta^2 \sqrt{\alpha}}{\delta^2} \). Hence \( D^2 : \Delta^2 = d : \delta \), and \( \frac{\sqrt{a}}{\sqrt{b}} = \frac{\Delta}{\delta} \), that is, the squares of the diameters of the horizontal tubes ought to be to one another in the compound ratio of the squares of the diameters of the adjutages, and the square roots of the altitudes of the reservoir. Now, it appears from the experiments of Mariotte (Traité du Mouvement des Eaux), that when the altitude of the reservoir is 16 feet, and the diameter of the adjutage six lines, the diameter of the horizontal tube ought to be 28 lines and a half. By taking this as a standard, therefore, the diameters of the horizontal tube may be easily found by the preceding rule, whatever be the altitude of the reservoir and the diameter of the adjutage.

It results from the three last experiments, that the jets rise to the smaller height when the adjutage is a cylindrical tube (see D, fig. 71.), that a conical adjutage throws the fluid very much higher, and that when the adjutage is a simple orifice the jet rises highest of all.

239. By comparing the preceding experiments with those of Mariotte, it appears, that the differences between the heights of vertical jets, and the heights of the reservoir, are nearly as the squares of the heights of the jets. Thus, \( a : b : c : d = E : b^2 : F : d^2 \); therefore, if \( a \) be known by experiment, we shall have \( c d = \frac{a b \times F d^2}{E b^2} \), and by adding \( e d \) to \( F d \), we shall have the altitude of the reservoir. But if \( F c \) were given, and it were required to find \( F d \), the height of the jet, we have, by the preceding analogy, \( F d^2 = \frac{E b^2 \times c d}{a b} \). But \( c d \) is an unknown quantity, and is equal to \( F c - F d \), therefore, by substitution, \( F d^2 = \frac{E b^2 \times F c - F d}{a b} \), or \( F d^2 \times \frac{E b^2}{a b} \times F d = \frac{E b^2 \times F c}{a b} \), which is evidently a quadratic equation, which, after reduction, becomes \( F d = \sqrt{\frac{E b^2 \times F c}{a b} + \frac{E b^4}{4} - \frac{E b^2}{2}} \).

240. From a comparison of the 5th and 6th columns of the table, it appears that a small inclination of the jet, to elimination of a vertical line, makes it rise higher than when it ascends the jet in exactly vertical; but even then it still falls short of the height of the reservoir. When the water first escapes from the adjutage, it generally springs higher than the reservoir; but this effect is merely momentary, as the jet instantly subsides, and continues at the altitudes exhibited in the foregoing tables. The great size of the jet at its first formation, and its subsequent diminution, have been ascribed to the elasticity of the air which follows the water in its passage through the orifice; but it is obvious, that this air, which moves along with the fluid, can never give it an impulsive force. In order to explain this phenomenon, let us suppose the adjutage to be stopped; then the air which the water drags along with it, will lodge itself at the extremity of the adjutage, so that there will be no water contiguous to the body which covers the orifice. As soon as the cover is removed from the adjutage, the imprisoned air escapes; the water immediately behind it rushes into the space which it leaves, and thus acquires in the tube a certain velocity which increases at the orifice in the ratio of the area of the section of the tube to the area of the section of the orifice (art. 189, note). When the orifice is small in comparison with the tube, the velocity of the issuing fluid must be considerable, and will raise it higher than the reservoir. But as the jet is resisted by the air, and retarded by the descending fluid, its altitude diminishes, and the simple pressure of the fluid becomes the only permanent source of its velocity. The preceding phenomenon was first noticed by Toricelli, who seems to ascribe the diminution in the altitude of the jet to the gravity of the descending particles.

241. The following table exhibits all that is necessary in the formation of jets. The two first columns are taken from Mariotte, and shew the altitude of the reservoir requisite to producing a jet of a certain height. The third column contains, in Paris pints, 36 of which are equal to a cubic foot, the quantity of water discharged in a minute by an orifice six lines in diameter. The fourth column, computed from the hypothesis in art. 238 contains the diameters of the horizontal tubes for an adjutage six lines in diameter, relative to the altitudes in the second column. The thickness of the horizontal tubes will be determined in a subsequent section.

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1 This was also observed by Wolfius, Opera Mathematica, tom. i. p. 302, schol. iv. 2 De Motu Projectorum. Oper. Geometr. p. 192. 3 Traité du Mouvement des Eaux, part iv. disc. 1. p. 303. 242. We have already seen that jets do not rise to the heights of their reservoirs; and have remarked that the difference between theory and experiment arises from the friction at the orifice, and the resistance of the air. The diminution of velocity produced by friction is very small, and the resistance of the air is a very inconsiderable source of retardation, unless when the jet rises to a great altitude. We must seek, therefore, for another cause of obstruction to the rising jet, which, when combined with these, may be adequate to the effect produced. Wolfius\(^1\) has very properly ascribed the diminution in the altitude of the jet to the gravity of the falling water. When the velocity of the foremost particles is completely spent, those immediately behind, by impinging against them, lose their velocity, and, in consequence of this constant struggle between the ascending and descending fluid, the jet continues at an altitude less than that of the reservoir. Hence we may discover the reason why an inclination of the jet increases its altitude; for the descending fluid falling a little to one side does not encounter the rising particles, and therefore permits them to reach a greater altitude than when their ascension is in a vertical line. Wolfius observes, in proof of his remark that the diminution is occasioned also by the weight of the ascending fluid, that mercury rises to a less height than water; but this cannot be owing to the greater specific gravity of mercury; for though the weight of the mercurial particles is greater than that of water, yet the momentum with which they ascend is proportionally greater, and therefore the resistance which opposes their tendency downwards, has the same relation to their gravity, as the resistance in the case of water has to the weight of the aqueous particles.

243. The theory of oblique jets has already been discussed in Prop. IX. art. 200. The two following experiments are given by Bossut. When the height NS of the reservoir AB (fig. 72) was 9 feet, and the diameter of the adjutage at N, 6 lines, a vertical abscissa CN of 4 feet 3 inches and 7 lines, answered to a horizontal ordinate CT of 11 feet 3 inches and 3 lines. When the altitude NS of the reservoir was 4 feet, the adjutage remaining the same, a vertical abscissa CN of 4 feet 3 inches and 7 lines, corresponded with a horizontal ordinate CT of 8 feet 2 inches and 8 lines. The real amplitudes, therefore, are less than those deduced from theory; and both are very nearly as the square roots of the altitudes of the reservoirs. Hence, to find the amplitude of a jet when the height of the reservoir is 10 feet, and the vertical abscissa the same, we have \( \sqrt{9} \text{ feet} : \sqrt{16} \text{ feet} = 11 \text{ feet } 3 \text{ inches } 3 \text{ lines} : 15 \text{ feet } 4 \text{ lines} \), the amplitude of the jet required. This rule, however, will apply only to small reservoirs; for when the jets enlarge, the curve which they describe cannot be determined by theory, and therefore the relation between the amplitudes and the heights of the reservoirs must be uncertain.

244. The following experiments on oblique jets were performed by MM. Michelotti and Venturi. When the height of the jet and the adjutage above a horizontal plane was 19.33 inches, the Venturi amplitude of projection, according to Michelotti, was 23.2 inches, with a simple orifice, and 20 inches with an additional tube. Venturi found, that when the height of the water in the reservoir was 32\(\frac{1}{2}\) inches, and that of the adjutage above a horizontal plane 54 inches, the amplitude of projection was 81\(\frac{1}{2}\) inches with a simple orifice, and 69 inches with an additional tube.

Sect. V. Experiments on the Motion of Water in Conduit Pipes.

245. The experiments of the Chevalier de Busat will be given at great length in the article Water-Works. That water is valuable on a subject of such public importance, we shall at present give a concise view of the experiments of Couplet, Bossut, and Prony, and of the practical conclusions which they authorize us to form.

246. It must be evident to every reader, that, when water is conducted from a reservoir by means of a long horizontal pipe, the velocity with which the water enters the pipe will be much greater than the velocity with which it issues from its farther extremity; and that, if the pipe has various flexures or bendings, the velocity with which the water leaves the pipe will be still farther diminished. The difference, therefore, between the initial velocity of the water, and the velocity with which it issues, will increase with the length of the pipe and the number of its flexures. By means of the theory, corrected by the preceding experiments, it is easy to determine with great accuracy the initial velocity of the water, or that with which it enters the pipe; but on the obstructions which the fluid experiences in its progress through the pipe, and on the causes of these obstructions, theory throws but a feeble light. The experiments of Bossut afford much instruction.

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\(^1\) Wolfius Opera Mathematica, tom. I. p. 802, schol. 4. ### Table XIII

**Containing the Quantities of Water discharged by Conduit Pipes of different lengths and diameters, compared with the Quantities discharged from additional tubes inserted in the same Reservoir.**

| Constant altitude of the water in the reservoir above the axis of the tube | Length of the conduit pipes | Quantity of water discharged in a minute by an additional tube | Quantity of water discharged by the conduit pipe in a minute | Ratio between the quantities of water furnished by the tube and the pipe of 16 lines diameter | Quantity of water discharged by an additional tube in a minute | Quantity of water discharged by the conduit pipe in a minute | Ratio between the quantities of water furnished by the tube and the pipe of 24 lines diameter | |---|---|---|---|---|---|---|---| | Feet | Feet | Cubic Inches | Cubic Inches | Tube and pipe 16 lines diam. | Tube and pipe 24 lines diam. | | 1 | 30 | 6330 | 2778 | 1 to .4389 | 14243 | 7680 | 1 to .5392 | | 1 | 60 | 6330 | 1957 | 1 to .3091 | 14243 | 5564 | 1 to .3906 | | 1 | 90 | 6330 | 1587 | 1 to .2507 | 14243 | 4534 | 1 to .3183 | | 1 | 120 | 6330 | 1351 | 1 to .2134 | 14243 | 3944 | 1 to .2769 | | 1 | 150 | 6330 | 1178 | 1 to .1861 | 14243 | 3486 | 1 to .2448 | | 1 | 180 | 6330 | 1052 | 1 to .1652 | 14243 | 3119 | 1 to .2190 | | 2 | 30 | 8939 | 4066 | 1 to .4548 | 20112 | 11219 | 1 to .5578 | | 2 | 60 | 8939 | 2888 | 1 to .3231 | 20112 | 8190 | 1 to .4072 | | 2 | 90 | 8939 | 2352 | 1 to .2631 | 20112 | 6812 | 1 to .3387 | | 2 | 120 | 8939 | 2011 | 1 to .2250 | 20112 | 5885 | 1 to .2926 | | 2 | 150 | 8939 | 1762 | 1 to .1971 | 20112 | 5232 | 1 to .2601 | | 2 | 180 | 8939 | 1583 | 1 to .1770 | 20112 | 4710 | 1 to .2341 |

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247. The third column of the preceding table contains the quantity of water discharged through an additional cylindrical tube 16 lines in diameter, or the quantity discharged from the reservoir into a conduit pipe of the same diameter; and the fourth column contains the quantity discharged by the conduit pipe. The fifth column, therefore, which contains the ratio between these quantities, will also contain the ratio between the velocity of the water at its entrance into the conduit pipe, which we shall afterwards call its initial velocity, and its velocity when it issues from the pipe, which shall be denominated its final velocity; for the velocities are as the quantities discharged, when the orifices are the same. The same may be said of the 6th, 7th, and 8th columns, with this difference only, that they apply to a cylindrical tube and a conduit pipe 24 lines in diameter.

248. By examining some of the experiments in the foregoing table, it will appear that the water sometimes loses half of its initial velocity. The velocity thus lost is consumed by the friction of the water on the sides of the pipe, as the quantities discharged, and consequently the velocities diminish when the length of the pipe is increased. In simple orifices, the friction is in the inverse ratio of their diameter; and it appears from the table, that the velocity of the water is more retarded in the pipe 16 lines in diameter, than in the other, which has a diameter of 24 lines. But though the velocity decreases when the length of the tube is increased, it by no means decreases in a regular arithmetical progression, as some authors have maintained. This is obvious from the table, from which it appears, that the differences between the quantities discharged, which represent also the differences between the velocities, always decrease, whereas the differences would have been equal, had the velocities decreased in an arithmetical progression. The same truth is capable of a physical explanation. If every filament of the fluid rubbed against the sides of the conduit pipe, then, since in equal times they all experience the same degree of friction, the velocities must diminish in the direct ratio of the lengths of the tubes, and will form a regular arithmetical progression, of which the first term will be the final, and the last the initial velocity of the water. But it is only the lateral filaments that are exposed to friction. This retards their motion; and the adjacent filaments which do not touch the pipe, by the adhesion to those which do touch it, experience also a retardation, but in a less degree, and go on with the rest, each filament sustaining a diminution of velocity inversely proportional to its distance from the sides of the pipe. The lateral filaments alone, therefore, provided they always remain in contact with the sides of the pipe, will have their velocities diminished in arithmetical progression, while the velocities of the central filaments will not decrease in a much slower progression; consequently, the mean velocity of the fluid, or that to which the quantities discharged are proportional, will decrease less rapidly than the terms of an arithmetical progression.

249. When the altitude of the reservoir was two feet, the retardation of discharge, and consequently of velocity, diminishes as the square roots of the altitudes; therefore friction must also be as the square roots of the altitudes of the reservoir. On some occasions Coulomb found that the friction of solid bodies diminished with an augmentation of velocity, but there is no ground for supposing that this takes place in the case of fluids.

250. When the pipe is inclined to the horizon, as CGF, the water will move with a greater velocity than in the horizontal tube CG h f. In the former case, the relative gravity of the water, which is to its absolute gravity as F increased to C f, or as the height of the inclined plane to its length, by its relative gravity accelerates its motion along the tube. But this acceleration takes place only when the inclination is considerable; for if the angle which the direction of the pipe forms with the horizon were no more than one degree, the retardation of friction would completely counterbalance the accel- Experiments on the Motion of Fluids.

Thus when the pipe CF, 16 lines in diameter, was 177 feet, and was divided into three equal parts in the points D and E, so that CD was 59 feet, CE 118 feet; and when CF was to F as 2124 to 241, the quantity of water discharged at F was 5795 cubic inches in a minute, the quantity discharged at E was 5801 cubic inches in a minute, and the quantity at D 5808 cubic inches. The quantities discharged therefore, and consequently the velocities, decreased from C to F; whereas if there had been no friction, and no adhesion between the aqueous particles, the velocities would have increased along the line of Vena CF in the subduplicate ratio of the altitudes CB, D m, E n, and F o; AB being the surface of the water in the reservoir. The preceding numbers, representing the quantities discharged at FE and D, decrease very slowly; consequently, by increasing the relative gravity of the water, that is, by inclining the tube more to the horizon, the effects of friction may be exactly counterbalanced. This inclination happens when the angle f CF is about 6° 31', or when F' of the pipe is the eighth or ninth part of CF. The quantities discharged at CDE and F will then be equal, and friction will have consumed the velocity arising from the relative gravity of the included water.

In order to determine the effects produced by flexures or sinuosities in conduit pipes, M. Bossut made the following experiments.

**Table XIV. Shewing the Quantities of Water discharged by rectilineal and curvilineal leaden Pipes, 50 feet long, and 1 inch in diameter.**

| Altitude of the Water in the Reservoir | Form of the Conduit Pipes.—See Figures 74. and 75. | Quantities of Water discharged in a Minute | |---------------------------------------|-------------------------------------------------|------------------------------------------| | Feet. Inches | | Cubic Inches | | 0 4 | The rectilineal tube MN placed horizontally, | 576 | | 1 0 | The same tube similarly placed, | 1050 | | 0 4 | The same tube bent into the curvilineal form ABC, fig. 74, each flexure lying flat on a horizontal plane, ABC being a horizontal section, | 540 | | 1 0 | The same tube similarly placed, | 1030 | | 0 4 | The same tube placed as in fig. 75, where ABCD is a vertical section, the parts A, B, C, D, rising above a horizontal plane, and the parts a, b, c, lying upon it, | 520 | | 1 0 | The same tube similarly placed, | 1028 |

Fig. 74. Fig. 75.

1. The two first experiments of the preceding table shew, that the quantities of water discharged diminish as the altitude of the reservoir. This arises from an increase of velocity, which produces an increase of friction.

2. The four first experiments shew, that a curvilineal pipe, in which the flexures lie horizontally, discharges less water than a rectilineal pipe of the same length. The friction being the same in both cases, this difference must arise from the impulse of the fluid against the angles of the tube; for if the tube formed an accurate curve, it is demonstrable that the curvature would not diminish the velocity of the water.

3. By comparing the 1st and 5th, and the 2d and 6th, experiments, it appears that, when the flexures are vertical, the quantity discharged is diminished. This also arises from the imperfection of curvature.

4. It appears, from a comparison of the 3d and 5th, with the 4th and 6th experiments, that when the flexures are vertical, the quantity discharged is less than when they are horizontal. In the former case, the motion of the fluid arises from the central impulsion of the water, retarded by its gravity in the ascending parts of the pipe, and accelerated in the descending parts; whereas the motion, in the latter case, arises wholly from the central impulsion of the fluid. To these points of difference the diminution of velocity may somehow or other be owing.

When a large pipe has a number of contrary flexures, the air sometimes mixes with the water, and occupies the highest parts of each flexure, as at B and C, fig. 75. By Fig. 75, this means the velocity of the fluid is greatly retarded, and the quantities discharged much diminished. This ought to be prevented by placing small tubes at B and C, having a small valve at their top.

A set of valuable experiments, on a large scale, were made by M. Couplet upon the motion of water in conduits pipes, and a detailed account of them is given in the Memoirs of the Academy for 1732, in his paper entitled *Des Recherches sur le Mouvement des Eaux dans les Tuyaux de conduite*. These experiments are combined with those of the Abbé Bossut in the following table, which gives a distinct view of all that they have done on this subject, and will be of great use to the practical engineer. ### Table XV. Containing the Results of the Experiments of Couplet and Bossut on Conduit Pipes differing in form, length, diameter, and in the materials of which they are composed,—under different Altitudes of water in the Reservoir.

| Altitude of the Water in the Reservoir | Length of the Conduit Pipe | Diameter of the Conduit Pipes | Nature, Position, and Form of the Conduit Pipes | Ratio between the Quantities which would be discharged if the Fluid experienced no resistance in the pipes, and the Quantities actually discharged—or the Ratio between the initial and the final Velocities of the Fluid. | |----------------------------------------|---------------------------|-------------------------------|-------------------------------------------------|------------------------------------------------------------------| | Ft. In. Lin. Feet. Lines. | | | | | | 0 4 0 50 12 | Rectilineal and horizontal pipe of lead, | 1 to 0.281 | | 1 0 0 50 12 | The same pipe similarly placed, | 1 to 0.305 | | 0 4 0 50 12 | The same pipe with several horizontal flexures, | 1 to 0.264 | | 1 0 0 50 12 | Same pipe, | 1 to 0.291 | | 0 4 0 50 12 | The same pipe with several vertical flexures, | 1 to 0.254 | | 1 0 0 50 12 | Same pipe, | 1 to 0.290 | | 1 0 0 180 16 | Rectilineal and horizontal pipe of white iron, | 1 to 0.166 | | 2 0 0 180 16 | Same pipe, | 1 to 0.177 | | 1 0 0 180 24 | Rectilineal and horizontal pipe of white iron, | 1 to 0.218 | | 2 0 0 180 24 | Same pipe, | 1 to 0.234 | | 20 11 0 177 16 | Rectilineal pipe of white iron, and inclined so that CF (fig. 73.) is to F as 2124 is to 241. | 1 to 0.2000 | | 13 4 8 118 16 | Rectilineal pipe of white iron, and inclined like the last, | 1 to 0.2500 | | 6 8 4 159 16 | Rectilineal pipe of white iron, and inclined like the last, | 1 to 0.354 | | 0 9 0 1782 48 | Conduit pipe almost entirely of iron, with several flexures both horizontal and vertical, | 1 to 0.350 | | 1 9 0 1782 48 | Same pipe, | 1 to 0.0376 | | 2 7 0 1782 48 | Same pipe, | 1 to 0.0387 | | 0 3 0 1710 72 | Conduit pipe almost entirely of iron, with several flexures both horizontal and vertical, | 1 to 0.0809 | | 0 5 3 1710 72 | Same pipe, | 1 to 0.0878 | | 0 5 7 7020 60 | Conduit pipe, partly stone and partly lead, with several flexures both horizontal and vertical, | 1 to 0.0432 | | 0 11 4 7020 60 | Same pipe, | 1 to 0.0476 | | 1 4 9 7020 60 | Same pipe, | 1 to 0.0513 | | 1 9 1 7020 60 | Same pipe, | 1 to 0.0532 | | 2 1 0 7020 60 | Same pipe, | 1 to 0.0541 | | 12 1 3 3600 144 | Conduit pipe of iron, with flexures both horizontal and vertical, | 1 to 0.0992 | | 12 1 3 3600 216 | Conduit pipe of iron, with several flexures both horizontal and vertical, | 1 to 0.1653 | | 4 7 6 4740 216 | Conduit pipe of iron, with several flexures both horizontal and vertical, | 1 to 0.0989 | | 20 3 0 14040 144 | Conduit pipe of iron, with several flexures both horizontal and vertical, | 1 to 0.0517 |

254. In order to show the application of the preceding results, let us suppose that a spring, or a number of springs combined, furnishes 40,000 cubic inches of water in one minute; and that it is required to conduct it to a given place 4 feet below the level of the spring, and so situated that the length of the pipe must be 2400 feet. It appears from Table VI. art. 226, that the quantity of water furnished in a minute by a short cylindrical tube, when the altitude of the fluid in the reservoir is 4 feet, is 7070 cubic inches; and since the quantities furnished by two cylindrical pipes under the same altitude of water are as the squares of their diameters, we shall have by the following analogy the diameter of the tube necessary for discharging 40,000 cubic inches in a minute: \( \sqrt{70720} : \sqrt{40000} = 12 \) lines or 1 inch : 28.4 lines, the diameter required. But by comparing some of the experiments in the preceding table, it appears that, when the length of the pipe is nearly 2400 feet, it will admit only about one-eighth of the water, that is, about 5000 cubic inches. That the pipe, however, may transmit the whole 40,000 cubic inches, its diameter must be increased. The following analogy, therefore, will furnish us with this new diameter: \( \sqrt{5000} : \sqrt{40000} = 28.54 \) lines : 80.73 lines, or 6 inches 8.5 lines, the diameter of the pipe which will discharge 40,000 cubic inches of water when its length is 2400 feet.

255. The following experiments were made by M. Bossut of Mezières in October 1779; and they are highly interesting, as they were made on the water discharged from the public and private fountains of that city. Table XVI. Containing Bossut's Experiments on the Quantities of Water discharged by different Pipes of various Lengths, and with different Adjutages, at the public and private Fountains of Mezieres.

| Head of Water | Length of Pipe | Diameter of Pipe | Size of Orifice | Ratio of the Real to the Theoretical Discharges | Ratio of the Height due to the Velocity to the Head of Water | Cubic Inches of Water discharged in a Minute | |---------------|----------------|------------------|----------------|-----------------------------------------------|-------------------------------------------------|---------------------------------------------| | Feet. In. | Feet. | Lines. | Lines. | | | | | 24 7 | 161 | 12 | 7½ | 0.045 | 0.002 | 242 | | 23 9 | 192 | 12 | 5½ | 0.075 | 0.006 | 230 | | 19 3 | 193 | 12 | 6½ | 0.068 | 0.005 | 222 | | 19 9 | 188 | 12 | 6½ | 0.061 | 0.004 | 237 | | 19 10 | 146 | 12 | 2½ by 7 | 0.089 | 0.008 | 168 | | 29 1 | 187 | 15 | 7½ by 5½ | 0.105 | 0.011 | 588 | | 8 0 | 1069 | 18 | Two adjutages, each 6 lines | 0.435 | 0.189 | 1686 | | 24 7 | 278 | 15 | 3½ | 0.396 | 0.157 | 458 | | 32 7 | 314 | 15 | Two adjutages, having each 5 lines | 0.227 | 0.052 | 1232 | | 30 5 | 446 | 18 | 2 by 6½ | 0.037 | 0.001 | 636 | | 26 3 | 506 | 18 | 4 | 0.447 | 0.200 | 696 | | 27 0 | 668 | 18 | 5½ | 0.301 | 0.091 | 900 | | 30 0 | 812 | 18 | 11 | 0.048 | 0.002 | 600 | | 10 5 | 194 | 12 | 5 | 0.377 | 0.139 | 576 | | 10 11 | 462 | 12 | 5½ | 0.332 | 0.109 | 576 | | 10 0 | 420 | 15 | 7 | 0.163 | 0.028 | 483 |

Sect. VI. Experiments on the Pressure exerted upon Pipes by the water which flows through them.

256. The pressure exerted upon the sides of conduit pipes by the included water has been already investigated theoretically in Prop. X. Part II. The only way of ascertaining by experiment the magnitude of this lateral pressure is to make an orifice in the side of the pipe, and find the quantity of water which it discharges in a given time.

This lateral pressure is the force which impels the water through the orifice; and therefore the quantity discharged, or the effect produced, must be always proportional to that pressure as its producing cause, and may be employed to represent it. The following table, founded on the experiments of Bossut, contains the quantities of water discharged from a lateral orifice about 3½ lines in diameter, according to theory and experiment.

Table XVII. Containing the Quantities discharged by a Lateral Orifice, or the Pressures on the Sides of Pipes, according to Theory and Experiment.

| Altitude of the Water in the Reservoir | Length of the Conduit Pipe | Quantities of Water discharged in 1 Minute, according to Theory | Quantities of Water discharged in 1 Minute, according to Experiment | |----------------------------------------|----------------------------|---------------------------------------------------------------|---------------------------------------------------------------| | Feet. | Feet. | Cubic Inches. | Cubic Inches. | | 1 | 30 | 176 | 171 | | 1 | 60 | 186 | 186 | | 1 | 90 | 190 | 190 | | 1 | 120 | 191 | 191 | | 1 | 150 | 192 | 193 | | 1 | 180 | 193 | 194 | | 2 | 30 | 244 | 240 | | 2 | 60 | 259 | 256 | | 2 | 90 | 264 | 261 | | 2 | 120 | 267 | 264 | | 2 | 150 | 268 | 265 | | 2 | 180 | 269 | 266 |

It appears from the preceding table, that the real lateral pressure in conduit pipes differs very little from that which is computed from the formula; but in order that this accordance may take place, the orifice must be so perforated, that its circumference is exactly perpendicular to the direction of the water, otherwise a portion of the water dis- 257. As pipes are exposed to forces besides those arising from the included water, they must be made much stronger than the preceding experiments would seem to require. The thicknesses of iron and leaden pipes used in France in the time of Bossut, are given in the following table.

| Iron Pipes | Leaden Pipes | |------------|-------------| | Diameter | Thickness | Diameter | Thickness | | Inches | Lines | Inches | Lines | | 1 | 1 | 1 | 2½ | | 2 | 3 | 1½ | 3 | | 4 | 4 | 2 | 4 | | 6 | 5 | 3 | 5 | | 8 | 6 | 4½ | 6 | | 10 | 7 | 6 | 7 | | 12 | 8 | 7 | 8 |

Sect. VII. Experiments on the Motion of Water in Canals.

258. Among the numerous experiments which have been made on this important subject, those of the Abbé Bossut seem entitled to the greatest confidence. His experiments were made on a rectangular canal 105 feet long, 5 inches broad at the bottom, and from 8 to 9 inches deep. The velocity of the water which transmitted the water from the reservoir into the canal was rectangular, having its horizontal base constantly five inches, and its vertical height sometimes half an inch, and at other times an inch. The sides of this orifice were made of copper, and rising perpendicularly from the side of the reservoir, they formed two vertical planes parallel to each other. This projecting orifice was fitted into the canal, which was divided into 5 equal parts of 21 feet each, and also into 3 equal parts of 35, and the time was noted which the water employed in reaching these points of division. The arrival of the water at these points was indicated by the motion of a very small water wheel placed at each, and impelled by the stream. When the canal was horizontal, the following results were obtained.

Table XVIII. Containing the Velocity of Water in a Rectangular Horizontal Canal 105 feet long, under different Altitudes of Fluid in the Reservoir.

| Altitude of the water in the reservoir | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Space run through by the water. | |---------------------------------------|---------|---------|---------|---------|---------|---------|---------|-------------------------------| | Vertical breadth of the orifice. | ½ an inch. | ½ an inch. | ½ an inch. | ½ an inch. | ½ an inch. | ½ an inch. | ½ an inch. | Feet. | | Time in which the number of feet in column seventh are run through by the water. | 2½ | 3½ | 3½+ | 2½+ | 2½+ | 3½- | 21 | | 5 | 7 | 9 | 4 | 5 | 6½ | 42 | | 10 | 13 | 17+ | 7 | 9 | 11+ | 63 | | 16 | 20 | 27+ | 11 | 14 | 18+ | 84 | | 23½ | 28½ | 38+ | 16½ | 20 | 26 | 105 |

259. It appears from column 1st, that the times successively employed to run through spaces of 21 feet each, are as the numbers 2, 3½, 5, 6, 7½, which form nearly an arithmetical progression, whose terms differ nearly by 1, so that by continuing the progression we may determine very nearly the time in which the fluid would run through any number of feet not contained in the 7th column. The same may be done with the other columns of the table.

If we compute theoretically the time which the water should employ in running through the whole length of the canal, or 105 feet, we shall find, that under the circumstances for each column of the preceding table the times, reckoning from the first column, are 6½, 3½, 7½, 8¾, 11½, 3¾, 6½, 3½, 7½, 8¾, 11½. It appears, therefore, by comparing these times with those found by experiment, that the velocity of the stream is very much retarded by friction, and that this retardation is less as the breadth of the orifice is increased; for since a greater quantity of water issues in this case from the reservoir, it has more power to overcome the obstacles which obstruct its progress. The signs + and — affixed to the numbers in the preceding table, indicate that these numbers are a little too great or too small.

260. The following experiments were made on inclined canals with different declivities, and will be of great use to the engineer. The inclination of the canal is the vertical distance of one of its extremities from a horizontal line which passes through its other extremity. ### Table XIX. Containing the Velocity of Water in a Rectangular Inclined Canal 105 Feet long, and under different Altitudes of Fluid in the Reservoir.

| Altitude of water in the reservoir | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | |-----------------------------------|---------|---------|---------|---------|---------|---------|---------|---------| | Inclination of the canal. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | | Height of the orifice ¼ an inch. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | | Inclination of the canal. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | | Height of the orifice 1 inch. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | | Inclination of the canal. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | | Height of the orifice 1 inch. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | | Inclination of the canal. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | | Height of the orifice 1 inch. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | | Inclination of the canal. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | | Height of the orifice 1½ inch. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. | Ft. In. |

Time in which the number of feet in the last col. is run through by the water.

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261. In the preceding experiments, the velocity of the first portion of water that issues from the reservoir was only observed; but when the current is once established, and its velocity permanent, it moves with greater rapidity, and there is always a fixed proportion between the velocity of the first portion of water and the permanent velocity of the established current. The cause of this difference Bossut does not seem to have thoroughly comprehended, when he ascribes it to a diminution of friction when the velocity becomes permanent. The velocity of the first portion of water that issues from the reservoir was measured by its arrival at certain divisions of the canal, consequently the velocity thus determined was the mean velocity of the water. The velocity of the established current, on the contrary, was measured by light bodies floating upon its surface, at the centre of the canal, therefore the velocity thus determined was the superficial velocity of the stream. But other the velocity of the superficial central filaments must be the greatest of all, because, being at the greatest distance from the sides and bottom of the canal, they are less affected by friction than any of the adjacent or inferior filaments, and are not retarded by the weight of any superincumbent fluid, and the superficial velocity of the current must, of consequence, be greater than its mean velocity, or, in other words, the velocity of the established current must exceed the velocity of the first portion of water. The following table contains the experiments of Bossut on this subject, the canal being of the same size as in the former experiments, but 600 feet long, and its inclination one-tenth of the whole, or 59.702 feet. ### Table XXI. Containing Du Buat's Experiments on the Motion of different Fluids, at different degrees of Temperature, in Tubes of Glass.

| Names of the Fluids | Diameter and length of the Pipe | Head of Water above the top of the tube | Height of the expense in a minute expressed in inches | Velocity in a second in inches | Degrees of Heat above the freezing point | |---------------------|---------------------------------|----------------------------------------|--------------------------------------------------|-------------------------------|------------------------------------------| | Rain water | | | | | | | Salt water | | | | | | | Salt water | Horizontal tube 2.9 | | | | | | Salt water | lines, or 0.24166 of an inch in diameter, and 36.25 inches long | | | | | | Alcohol | | | | | | | Mercury | | | | | | | Mercury | | | | | | | Mercury | | | | | | | Rain water | | | | | | | Rain water | | | | | | | Rain water | | | | | | | Rain water | Horizontal tube 2 | | | | | | Alcohol | | | | | | | Alcohol | lines, or 0.16666 of an inch in diameter, and 36.25 inches long | | | | | | Mercury | | | | | | | Mercury | | | | | | | Mercury | | | | | | | Alcohol | | | | | | | Alcohol | line in diameter, and 34.16666 inches long | | | | |

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**Sect. VIII. On the Influence of Heat on the Motion of Fluids.**

262. In all the experiments related in this chapter, and in those of the Chevalier Buat, which are given in the article *Water Works*, the temperature of the water employed has never been taken into consideration. That the fluidity of water is increased by heat can scarcely admit of a doubt. Professor Leslie, in his ingenious paper on Capillary Action, has proved by experiment that a jet of warm water will spring much higher than a jet of cold water, and hot considering that a siphon which discharges cold water only by drops; the temperature will discharge water of a high temperature in a continued stream. A similar fact was observed by the ancients, the water clock went slower in winter than in summer, and Warm water seems to attribute this retardation to a diminution of its fluidity. It is therefore obvious, that warm water will issue faster than from an aperture with greater velocity than cold water, and that the quantities of fluid discharged from the same orifice, and under the same pressure, will increase with the temperature of the fluid. Hence we may discover the cause of the great discrepancy between the experiments of different philosophers on the motion of fluids. Their experiments were performed in different climates and at different seasons of the year; and, as the temperature of the water would be variable from these and from other causes, a variation in their results was the inevitable consequence.

263. M. de Buat and M. Girard are the only persons who have made experiments on this interesting subject, and M. de Buat employed in his experiments tubes of a large diameter, and hence the effects of heat were not very conspicuous. The following table contains a general view of the results which he obtained:

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*Plutarch, Quæst. Natural.* Hence our author concludes that the velocity of water diminishes as its temperature approaches to that of the freezing point, and vice versa; that salt water has a less velocity than rain water, that alcohol runs slower than water, and mercury more rapidly.

264. The general result of M. Girard's experiments has been already given in the History of HYDRODYNAMICS. His experiments were made with copper tubes of exactly the same internal diameter, and drawn upon steel mandrills; and he employed two sets of these tubes of different diameters. The first set consisted of tubes, whose length was two decimeters, and diameter 2.96 millimetres, and they screwed into each other so as to form tubes of various lengths, from 20 to 222 centimetres. The second set consisted of smaller tubes, whose diameter was 1.83 millimetres. These tubes were then fixed horizontally in the sides of a reservoir, which was a cylinder of white iron 25 centimetres in diameter, and 5 decimetres high. The reservoir was kept full by the usual contrivances; and the water discharged by the tube subjected to trial, was received into a copper vessel horizontally, whose capacity had been accurately ascertained. The filling of the vessel was indicated by the instant when the water which it contained had wetted equally a plate of glass which covered almost the whole of its surface, and the time employed to fill this vessel was measured with great accuracy. The temperature of the water was also carefully noted. The results thus obtained amounted to 1200, and were arranged by M. Girard into thirty-four tables, according to the different circumstances of the experiment. When the capillary tube has such a length, that the term proportional to the square of the velocity disappears in the general formula, the velocity with which the fluid is discharged, is affected in a very singular manner by a variation of temperature. If the velocity is expressed by 10, when the temperature is 0° of the centigrade thermometer, the velocity will be so great as 42; or increased more than four times when the temperature amounts to 85° centigrade. When the length of the capillary tube is below the above mentioned limit, a variation of temperature exercises but a slight influence upon the velocity of the issuing fluid. If the length of the adutage, for example, is 55 millimetres, and if the velocity is represented by 10 at 5° of the centigrade thermometer, it will be represented only by 12 at a temperature of 87°. In conduit pipes of the ordinary diameter, a change of temperature produces almost no perceptible change in the velocity of efflux. M. Girard also found, that the quantity of water discharged by capillary tubes, varied not only with the fluids which were used, but with the nature of the solid substance of which the tubes were composed.

CHAPTER III.—ON THE RESISTANCE OF FLUIDS.

265. In the article Resistance of Fluids, the reader will find that important subject treated at great length. The researches of preceding philosophers are there given in full detail; their different theories are compared with experiments, and the defects of these minutely considered. Since that article was composed, this intricate subject has been investigated by other writers, and though they have not enriched the science of hydraulics with a legitimate theory of the resistance of fluids, the results of their labours cannot fail to be interesting to every philosopher.

266. The celebrated Coulomb has very successfully employed the principle of torsion, to determine the cohesion of fluids, and the laws of their resistance in very slow motions. His experiments are new, and were performed with the greatest accuracy; and the results which he obtained were perfectly conformable to the deductions of theory. We shall therefore endeavour to give the reader some idea of the discoveries which he has made.

267. When a body is struck by a fluid with a velocity exceeding eight or nine inches per second, the resistance has been found proportional to the square of the velocity, whether the body in motion strikes the fluid at rest, or the body is struck by the moving fluid. But when the velocity is so slow as not to exceed four-tenths of an inch in a second, the resistance is represented by two terms, one of which is proportional to the simple velocity, and the other to the square of the velocity. The first of these sources of resistance arises from the cohesion of the fluid particles which separate from one another, the number of particles thus separated being proportional to the velocity of the body. The other cause of resistance is the inertia of the particles, which, when struck by the fluid, acquire a certain degree of velocity proportional to the velocity of the body; and as the number of these particles is also proportional to that velocity, the resistance generated by their inertia must be proportional to the square of the velocity.

268. When Sir Isaac Newton\(^1\) was determining the resistance which the air opposed to the oscillatory motion of a globe in small oscillations, he employed a formula of three terms, one of them varying as the square of the velocity, the second as the \( \frac{3}{2} \) power of the velocity, and the third as the simple velocity; and in another part of the work he reduces the formula to two terms, one of which is as the square of the velocity, and the other constant. D. Bernoulli\(^2\) also supposes the resistance to be represented by two terms, one as the square of the velocity, and the other constant. M. Graveseude\(^3\) has found that the pressure of a fluid in motion against a body at rest, is partly proportional to the simple velocity, and partly to the square of the velocity. But when the body moves in a fluid at rest, he found\(^4\) the resistance proportional to the square of the velocity, and to a constant quantity. When the body in motion, therefore, meets the fluid at rest, these three philosophers have agreed, that the formula which represents the resistance of fluids consists of two terms, one of which is as the square of the velocity, and the other constant. The experiments of Coulomb, however, incontestably prove, that the pressure which the moving body in this case sustains, is represented by two terms, one proportional to the simple velocity, and the other to its square, and that if there is a constant quantity, it is so very small as to escape detection.

269. In order to apply the principle of torsion to the resistance of fluids, M. Coulomb made use of the apparatus represented in fig. 76. On the horizontal arm LK, which may be supported by a vertical stand, is fixed the small perforated circle fe, perforated in the centre, so as to admit the cylindrical pin ba. Into a slit in the extremity of this pin is fastened, by means of a screw, the brass wire ag, whose force of torsion is to be compared with the resistance of the fluid; and its lower extremity is fixed in the same way into a cylinder of copper gd, whose diameter is about four tenths of an inch. The cylinder gd is perpendicular to the disc DS, whose circumference is divided into 480 equal

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\(^1\) Principia, lib. ii. prop. xl.

\(^2\) Elements of Natural Philosophy, art. 1911.

\(^3\) Comment. Pretropol. tom. iii. and iv.

\(^4\) ib. art. 1975. RS is placed upon the point O, the zero of the circular scale. The small rule RM may be elevated or depressed at pleasure round its axis n, and the stand GH which supports it may be brought into any position round the horizontal disc. The lower extremity of the cylinder gd is immersed about two inches in the vessel of water MNOP, and to the extremity d is attached the planes, or the bodies whose resistance is to be determined when they oscillate in the fluid by the torsion of the brass wire. In order to produce these oscillations, the disc DS, supported by both hands, must be turned gently round to a certain distance from the index, without deranging the vertical position of the suspended wire. The disc is then left to itself; the force of torsion causes it to oscillate, and the successive diminutions of these oscillations are carefully observed. A simple formula gives in weights the force of torsion that produces the oscillations; and another formula well known to geometers, determines (by an approximation sufficiently accurate in practice), by means of the successive diminution of the oscillations compared with their amplitude, what is the law of the resistance, relative to the velocity, which produces those diminutions.

270. The method employed by Coulomb, in reducing his experiments, is similar to that adopted by Newton and other mathematicians, when they wished to determine the resistance of fluids, from the successive diminutions of the oscillations of a pendulum moving in a resisting medium; but is much better fitted for detecting the small quantities which are to be estimated in such researches. When the pendulum is employed, the specific gravity of the body, relative to that of the fluid, must be determined; and the least error in this point leads to very uncertain results. When the pendulum is in different points of the arc in which it oscillates, the wire or pendulum rod is plunged more or less in the fluid; and the alterations which may result from this are frequently more considerable than the small quantities which are the object of research. It is only in small oscillations, too, that the force which brings the pendulum from the vertical, is proportional to the angle which the pendulum rod, in different positions, forms with this vertical line; a condition which is necessary before the formulæ can be applied. But small oscillations are attended with great disadvantages; and their successive diminutions cannot be determined but by quantities which it is difficult to estimate exactly, and which are changed by the smallest motion either of the fluid in the vessel, or of the air in the chamber. In small velocities, the pendulum rod experiences a greater resistance at the point of floatation than at any other part. This resistance, too, is very changeable; for the water rises from its level along the resistance pendulum rod to greater or less heights, according to the velocity of the pendulum.

271. These and other inconveniences which might be mentioned, are so inseparable from the use of the pendulum, that Newton and Bernoulli have not been able to compare the laws of the resistance of fluids in very slow motions. When the resistance of fluids is compared with the force of torsion, these disadvantages do not exist. The force body is in this case entirely immersed in the fluid; and as of torsion every point of its surface oscillates in a horizontal plane, the relation between the densities of the fluid and the oscillating body has no influence whatever on the moving force. One or two circles of amplitude may be given to the oscillations; and their duration may be increased at pleasure, either by diminishing the diameter of the wire, or increasing its length; or, which may be more convenient, by augmenting the momentum of the horizontal disc. Coulomb, however, found that when each oscillation was so long as to continue about 100 seconds, the least motion of the fluid, or the tremor occasioned by the passing of a carriage, produced a sensible alteration on the results. The oscillations best fitted for experiments of this kind, continued from 20 to 30 seconds, and the amplitude of those that gave the most regular results, was comprehended between 480 degrees, the entire division of the disc, and 8 or 10 divisions reckoned from the zero of the scale. From these observations, it will be readily seen, that it is only in very slow motions that an oscillating body can be employed for determining the resistance of fluids. In small oscillations, or in quick circular motions, the fluid struck by the body is continually in motion; and when the oscillating body returns to its former position, its velocity is either increased or retarded by the motion communicated to the fluid, and not extinguished.

272. In the first set of experiments made by Coulomb, he attached to the lower extremity of the cylinder gd a velocity is circular plate of white iron, about 195 millimetres in diameter, and made it move so slowly, that the part of the resistance proportional to the square of the velocity wholly disappeared. For if, in any particular case, the portion of the resistance proportional to the simple velocity should be equal to the portion that is proportional to the square of the velocity when the body has a velocity of one-tenth of an inch per second, then, when the velocity is 100 times greater than that of an inch per second, the part proportional to the square of the velocity will be a hundred times greater than that proportional to the simple velocity; but if the velocity is only the hundredth part of the tenth of an inch per second, then the part proportional to the simple velocity will be 100 times greater than the part proportional to the square of the velocity.

273. When the oscillations of the white iron plate were so slow, that the part of the resistance which varies with Coulomb's second power of the velocity was greatly inferior to the other part, he found, from a variety of experiments, that the resistance which diminished the oscillations of the horizontal plate was uniformly proportional to the simple velocity, and that the other part of the resistance, which follows the ratio of the square of the velocity, produced no sensible change upon the motion of the white iron disc. He also found, in conformity with theory, that the momenta of resistance in different circular plates moving round their centre in a fluid, are as the fourth power of the diameters of these circles; and that, when a circle of 195 millimetres (6.677 English inches) in diameter, moved round its centre in water, so that its circumference had a velocity of 140 millimetres (5.512 English inches) per second, the momentum of resistance which the fluid opposed to its circular motion was equal to one-tenth of a gramme (1.544 English troy grains) placed at the end of a lever 143 millimetres (5.63 English inches) in length.

274. M. Coulomb repeated the same experiments in a vessel of clarified oil, at the temperature of 16 degrees of Reaumur. He found, as before, that the momenta of the resistance of different circular discs, moving round their centre in the plane of their superficies, were as the fourth power of their diameters; and that the difficulty with which the same horizontal plate, moving with the same velocity, separated the particles of oil, was to the difficulty with which it separated the particles of water, as 17.5 to 1, which is therefore the ratio that the mutual cohesion of the particles of oil has to the mutual cohesion of the particles of water.

275. In order to ascertain whether or not the resistance of a body moving in a fluid was influenced by the nature of its surface, M. Coulomb anointed the surface of the white iron plate with tallow, and wiped it partly away, so that the thickness of the plate might not be sensibly increased. The plate was then made to oscillate in water, and the oscillations were found to diminish in the same manner as before the application of the tallow. Over the surface of the tallow upon the plate, he afterwards scattered, by means of a sieve, a quantity of coarse sand which adhered to the greasy surface; but when the plate, thus prepared, was caused to oscillate, the augmentation of resistance was so small, that it could scarcely be appreciated. We may therefore conclude, that the part of the resistance which is proportional to the simple velocity, is owing to the mutual adhesion of the particles of the fluid, and not to the adhesion of these particles to the surface of the body.

276. If the part of the resistance varying with the simple velocity were increased when the white iron plate was immersed at greater depths in the water, we might suppose it to be owing to the friction of the water on the horizontal surface, which, like the friction of solid bodies, should be proportional to the superincumbent pressure. In order to settle this point, M. Coulomb made the white iron plate oscillate at the depth of two centimetres (0.787 English inches), and also at the depth of 50 centimetres (19.6855 English inches), and found no difference in the resistance; but as the surface of the water was loaded with the whole weight of the atmosphere, and as an additional load of 50 centimetres of water could scarcely produce a perceptible augmentation of the resistance, M. Coulomb employed another method of deciding the question. Having placed a vessel full of water under the receiver of an air-pump, the receiver being furnished with a rod and collar of leather at its top, he fixed to the hook, at the end of the rod, a harpsichord wire, number 7 in commerce, and suspended to it a cylinder of copper, like \( gd \), fig. 76, which plunged in the water of the vessel, and under this cylinder he fixed a circular plane, whose diameter was 101 millimetres (3.976 English inches). When the oscillations were finished, and consequently the force of torsion nothing, the zero of torsion was marked by the aid of an index fixed to the cylinder. The rod was then made to turn quickly round through a complete circle, which gave to the wire a complete circle of torsion, and the successive diminutions of the oscillations were carefully observed. The diminution for a complete circle of torsion was found to be nearly a fourth part of the circle for the first oscillation, but always the same whether the experiment was made in a vacuum or in the atmosphere. A small pallet 50 millimetres long (1.969 English inches), and 10 millimetres broad (0.3937 English inches), which struck the water perpendicular to its plane, furnished a similar result. We may therefore conclude, that when a submerged body moves in a fluid, the pressure which it sustains, measured by the altitude of the superior fluid, does not perceptibly increase the resistance; and consequently, that the part of this resistance proportional to the simple velocity, can in no respect be compared with the friction of solid bodies, which is always proportional to the resistance.

277. The next object of M. Coulomb was to ascertain the resistance experienced by cylinders that moved very slowly, and perpendicular to their axes; but as the particles of fluid struck by the cylinder necessarily partook of its moving motion, it was impossible to neglect the part of the resistance proportional to the square of the velocity, and therefore he was obliged to perform the experiments in such a manner that both parts of the resistance might be computed. The three cylinders which he employed were 249 millimetres (9.803 English inches) long. The first cylinder was 0.57 millimetres (0.0342 English inches or \( \frac{1}{3} \) of an inch) in circumference, the second 11.2 millimetres (0.4409 English inches), and the third 21.1 millimetres (.83807 English inches). They were fixed by their middle under the cylindrical piece \( dg \), so as to form two horizontal radii, the whose length was 124.5 millimetres (4.901 English inches) or half the length of each cylinder. After making the necessary experiments and computations, he found that the part of the resistance proportional to the simple velocity, which, to avoid circumlocution, we shall call \( r_1 \), did not therefore vary with the circumferences of the cylinders. The circumferences of the first and third cylinders were to one another as 24 : 1, whereas the resistances were in the ratio of 3 : 1. The same conclusion was deduced by comparing the experiments made with the first and second cylinder.

278. In order to explain these results, M. Coulomb very justly supposes, that, in consequence of the mutual adhesion of the particles of water, the motion of the cylinder is communicated to the particles at a small distance from it. The particles which touch the cylinder have the same velocity as the cylinder, those at a greater distance have a less velocity, and at the distance of about one-tenth of an inch the velocity ceases entirely, so that it is only at that distance from the cylinder that the mutual adhesion of the fluid molecules ceases to influence the resistance. The resistance \( r \) therefore should not be proportional to the circumference of the real cylinder, but to the circumference of a cylinder whose radius is greater than the real cylinder by one-tenth of an inch. It consequently becomes a matter of importance to determine with accuracy the quantity which must be added to the real cylinder in order to have the radius of the cylinder to which the resistance \( r \) is proportional, and from which it must be computed. Coulomb found the quantity by which the radius should be increased to be 1.5 millimetres (\( \frac{1}{60} \) of an English inch), so that the diameter of the augmented cylinder will exceed the diameter of the real cylinder by double that quantity, or \( \frac{1}{30} \) of an inch.

279. The part of the resistance varying with the square of the velocity, or that arising from the inertia of the fluid, since which we shall call \( R \), was likewise not proportional to the circumferences of the cylinder; but the augmentation of the radii amounts in this case only to \( \frac{1}{60} \) of an inch, portion which is only one-fifth of the augmentation necessary for the finding the resistance \( r \). The reason of this difference is manifest; all the particles of the fluid when they are separated from each other provide the same resistance, whatever be their velocity; consequently as the value of \( r \) depends only on the adhesion of the particles, the resistances are due to this adhesion will reach to the distance from the center of the cylinder where the velocity of the particles is 0. In comparing the different values of \( R \), the part of the resistance which varies as the square of the velocity, all the particles are supposed to have a velocity equal to that of the cylinder; but as it is only the particles which touch the cylinder that have this velocity, it follows that the augmentation of the diameter necessary for finding \( R \) must be less than the augmentation necessary for finding \( r \). 280. In determining experimentally the part of the momentum of resistance proportional to the velocity, by two cylinders of the same diameter, but of different lengths, M. Coulomb found that this momentum was proportional to the third power of their lengths. The same result may be deduced from theory; for supposing each cylinder divided into any number of parts, the length of each part will be proportional to the whole length. The velocity of the corresponding parts will be as these lengths, and also as the distance of the same parts from the centre of rotation. The theory likewise proves, that the momentum of resistance depending on the square of the velocity, in two cylinders of the same diameter but of different lengths, is proportional to the fourth power of the length of the cylinder.

281. When the cylinder 0.9803 inches in length, and 0.04409 inches in circumference, was made to oscillate in the fluid with a velocity of 5.51 inches per second, the part of the resistance \( r \) was equal to 58 milligrammes, or .8932 troy grains. And when the velocity was 0.3937 inches per second, the resistance \( r \) was 0.00414 grammes, or 0.637 troy grains.

282. The preceding experiments were also made in the oil formerly mentioned; and it likewise appeared, from their results, that the mutual adhesion of the particles of oil was to the mutual adhesion of the particles of water as 17 to 1. But though this be the case, M. Coulomb discovered that the quantity by which the radii of the cylinder must be augmented in order to have the resistance \( r \), is the very same as when the cylinder oscillated in water. This result was very unexpected, as the greater adhesion between the particles of oil might have led us to anticipate a much greater augmentation. When the cylinders oscillated both in oil and water with the same velocity, the part of the resistance \( R \) produced by the inertia of the fluid particles on the which the cylinder put in motion, was almost the same in both. As this part of the resistance depends on the quantity of particles put in motion, and not on their adhesion, the resistances due to the inertia of the particles will be in different fluids as their densities.

283. In a subsequent memoir, Coulomb proposes to determine numerically the part of the resistance proportional to the square of the velocity, and to ascertain the resistance extend his researches of globes with plain, convex, and concave surfaces. He has found in general that the resistance of bodies not entirely immersed in the fluid is much greater than that of fluids bodies which are wholly immersed; and he promises to make further experiments upon this point. We intended on the present occasion to have given the reader a more complete view of the researches of this ingenious philosopher; but these could not well be understood without a knowledge of his investigations respecting the force of torsion, which we have not yet had an opportunity of communicating. In the article Mechanics, however, we shall introduce the reader to this interesting subject; and may afterwards have an opportunity of making him farther acquainted with those researches of Coulomb, of which we have at present given only a general view.

284. The subject of the resistance of fluids has been recently treated by the learned Dr Hutton of Woolwich, of Dr Hut- His experiments were made in air with bodies of various forms, moving with different velocities, and inclined at various angles to the direction of their motion. The following table contains the results of many interesting experiments. The numbers in the ninth column represent the exponents of the power of the velocity which the resistances in the 8th column bear to each other.

| TABLE I. Shewing the Resistance of Hemispheres, Cones, Cylinders, and Globes, in different Positions, and moving with different Velocities. | | --- | --- | --- | --- | --- | --- | --- | --- | --- | | **Velocity per second.** | Small hemisphere, 4½ inches dia. flat side. | Large hemisphere 6½ inches diameter. | Cone 6½ inches diameter. | Cylinder 6½ inches diameter. | Globe 6½ inches diameter. | Power of the velocity to which the resistance is proportional. | | Feet. | Oz. avoird. | Oz. avoird. | Oz. avoird. | Oz. avoird. | Oz. avoird. | Oz. avoird. | Oz. avoird. | | 3 | .028 | .051 | .020 | .028 | .064 | .050 | .027 | | 4 | .048 | .096 | .039 | .048 | .109 | .060 | .047 | | 5 | .072 | .148 | .063 | .071 | .162 | .143 | .068 | | 6 | .103 | .211 | .092 | .098 | .225 | .205 | .094 | | 7 | .141 | .284 | .123 | .129 | .298 | .278 | .125 | | 8 | .184 | .368 | .160 | .168 | .382 | .360 | .162 | | 9 | .233 | .464 | .199 | .211 | .478 | .456 | .205 | | 10 | .287 | .573 | .242 | .260 | .587 | .565 | .255 | | 11 | .349 | .698 | .292 | .315 | .712 | .688 | .310 | 2.052 | | 12 | .418 | .836 | .347 | .376 | .850 | .826 | .370 | 2.042 | | 13 | .492 | .988 | .409 | .440 | 1.000 | .979 | .435 | 2.036 | | 14 | .573 | 1.154 | .478 | .512 | 1.166 | 1.145 | .505 | 2.031 | | 15 | .661 | 1.336 | .552 | .589 | 1.346 | 1.327 | .581 | 2.031 | | 16 | .754 | 1.538 | .634 | .673 | 1.546 | 1.526 | .663 | 2.033 | | 17 | .853 | 1.757 | .722 | .762 | 1.763 | 1.745 | .752 | 2.038 | | 18 | .959 | 1.998 | .818 | .858 | 2.002 | 1.986 | .848 | 2.044 | | 19 | 1.073 | 2.998 | .921 | .959 | 2.260 | 2.246 | .949 | 2.047 | | 20 | 1.196 | 2.542 | 1.033 | 1.069 | 2.540 | 2.528 | 1.057 | 2.051 |

Mean proportional numbers: 140, 288, 119, 126, 291, 285, 124, 2.040

285. From the preceding experiments we may draw the following conclusions: 1. That the resistance is nearly proportional to the surfaces, a small increase taking place when the surfaces and the velocities are great. 2. The resistance to the same surface moving with different velocities, is nearly as the square of the velocity; but it appears from On the 9th column that the exponent increases with the velocity. 3. The round and sharp ends of solids sustain a greater resistance than the flat ends of the same diameter. 4. The resistance to the base of the hemisphere is to the resistance on the convex side, or the whole sphere, as \(2\frac{1}{2}\) to 1, instead of 2 to 1, as given by theory. 5. The resistance on the base of the cone is to the resistance on the vertex nearly as \(2\frac{1}{2}\) to 1; and in the same ratio is radius to the sine of half the angle at the vertex. Hence in this case the resistance is directly as the sine of the angle of incidence, the transverse section being the same. 6. The resistance of the base of a hemisphere, the base of a cone, and the base of a cylinder, are all different, though these bases be exactly equal and similar.

286. The following table contains the resistance sustained by a globe 1.965 inches in diameter. The fourth column is the quotient of the resistance by experiment, divided by the theoretical resistance.

| Velocity of the Globe per second | Resistance by experiment | Resistance by theory | Ratio between the experimental and theoretical resistance | Power of the velocity to which the resistance is proportional | |---------------------------------|--------------------------|---------------------|--------------------------------------------------------|---------------------------------------------------------------| | Feet | Oz. avoird. | Oz. avoird. | | | | 5 | .0006 | .0005 | 1.20 | | | 10 | .0245 | .020 | 1.23 | | | 15 | .055 | .044 | 1.25 | | | 20 | .100 | .079 | 1.27 | | | 25 | .157 | .123 | 1.28 | 2.022 | | 30 | .23 | .177 | 1.30 | 2.059 | | 40 | .42 | .314 | 1.33 | 2.068 | | 50 | .67 | .491 | 1.36 | 2.075 | | 100 | 2.72 | 1.164 | 1.38 | 2.059 | | 200 | 11 | 7.9 | 1.40 | 2.041 | | 300 | 25 | 18.7 | 1.41 | 2.039 | | 400 | 45 | 31.4 | 1.43 | 2.039 | | 500 | 72 | 49 | 1.47 | 2.044 | | 600 | 107 | 71 | 1.51 | 2.051 | | 700 | 151 | 96 | 1.57 | 2.059 | | 800 | 205 | 126 | 1.63 | 2.067 | | 900 | 271 | 159 | 1.70 | 2.077 | | 1000 | 350 | 196 | 1.78 | 2.086 | | 1100 | 442 | 238 | 1.86 | 2.095 | | 1200 | 546 | 283 | 1.90 | 2.102 | | 1300 | 651 | 332 | 1.99 | 2.107 | | 1400 | 785 | 385 | 2.04 | 2.111 | | 1500 | 916 | 442 | 2.07 | 2.113 | | 1600 | 1051 | 503 | 2.09 | 2.113 | | 1700 | 1186 | 568 | 2.08 | 2.111 | | 1800 | 1319 | 636 | 2.07 | 2.108 | | 1900 | 1447 | 709 | 2.04 | 2.104 | | 2000 | 1569 | 786 | 2.00 | 2.098 |

287. It appears from a comparison of the 2d, 3d, and 4th columns, that when the velocity is small, the resistance by experiment is nearly equal to that deduced from theory; but that as the velocity increases, the former gradually exceeds the latter till the velocity is 1300 feet per second, when it becomes twice as great. The difference between the two resistances then increases, and reaches its maximum between the velocities of 1600 and 1700 feet. It afterwards decreases gradually as the velocity increases, and at the velocity of 2000 the resistance by experiment is again double of the theoretical resistance. By considering the numbers in column 5th, it will be seen, that in slow motions the resistances are nearly as the square of the velocities; that this ratio increases gradually, though not regularly, till at the velocity of 1500 or 1600 feet it arrives at its maximum. It then gradually diminishes as the velocity increases.

Conclusions similar to these were deduced from experiments made with globes of a larger size.

288. The following table contains the resistance of a plane inclined at various angles, according to experiment, and according to a formula deduced from the experiments.

**Table III. Containing the Resistances to a Plane Inclined at Various Angles to the Line of its Motion.**

| Degrees | Oz. avoird. | Oz. avoird. | Sines of the angles to radius .840 | |---------|-------------|-------------|-----------------------------------| | 0 | .000 | .000 | .000 | | 5 | .015 | .009 | .073 | | 10 | .044 | .035 | .146 | | 15 | .082 | .076 | .217 | | 20 | .133 | .131 | .287 | | 25 | .200 | .199 | .355 | | 30 | .278 | .278 | .420 | | 35 | .362 | .363 | .482 | | 40 | .448 | .450 | .540 | | 45 | .534 | .535 | .594 | | 50 | .619 | .613 | .643 | | 55 | .684 | .680 | .688 | | 60 | .729 | .736 | .727 | | 65 | .770 | .778 | .761 | | 70 | .803 | .808 | .789 | | 75 | .823 | .826 | .811 | | 80 | .835 | .836 | .827 | | 85 | .839 | .839 | .838 | | 90 | .840 | .840 | .840 |

289. The plane with which the preceding experiments were performed was 32 square inches, and always moved with a velocity of 12 feet per second. The resistances which this plane experienced are contained in column 2d. From the numbers in that column Dr Hutton deduced the formula \(.84s^3 + .84c\), where \(s\) is the sine, and \(c\) the cosine of the angles of inclination in the first column. The resistances computed from this formula are contained in column 3d, and agree very nearly with the resistances deduced from experiment. The 4th column contains the sines of the angles in the first column to a radius .84, in order to compare them with the resistances which have obviously no relation either to the sines of the angles or to any power of the sines. From the angle of 0 to about 60° the resistances are less than the sines; but from 60° to 90° they are somewhat greater.

290. The experiments of Mr Vince were made with bodies at a considerable depth below the surface of water; Mr Vince determined the resistance which they experienced, both when they moved in the fluid at rest, and when they received the impulse of the moving fluid. In the experiments contained in the following table, the body moved in the fluid with a velocity of 0.66 feet in a second. The angles at which the planes struck the fluid are contained in the first column. The second column shows the resist- The third column exhibits the resistance by theory, the perpendicular distance being supposed the same as by experiment. The fourth column shows the power of the sine of the angle to which the resistance is proportional, and was computed in the following manner. Let \( s \) be the sine of the angle, radius being 1; and \( r \) the resistance at that angle. Suppose \( r \) to vary as \( s^m \), then we have

\[ r^m : s^m = 0.2321 : r ; \text{ hence } s^m = \frac{1}{0.2321} \text{ and therefore } \]

\[ m = \frac{\log r - \log 0.2321}{\log s} \]

and by substituting their corresponding values, instead of \( r \) and \( s \) we shall have the values of \( m \) or the numbers in the fourth column.

**Table IV. Containing the resistance of a Plane Surface moving in a Fluid, and placed at different angles to the path of its motion.**

| Angle of inclination | Resistance by experiment | Resistance by theory | Power of the sine of the angle to which the resistance is proportional | |----------------------|--------------------------|---------------------|---------------------------------------------------------------| | Degrees | Troy ounces | Troy ounces | Experiments | | 10 | 0.0112 | 0.0012 | 1.73 | | 20 | 0.0364 | 0.0093 | 1.73 | | 30 | 0.0769 | 0.0290 | 1.54 | | 40 | 0.1174 | 0.0616 | 1.54 | | 50 | 0.1552 | 0.1043 | 1.51 | | 60 | 0.1902 | 0.1476 | 1.38 | | 70 | 0.2125 | 0.1926 | 1.42 | | 80 | 0.2237 | 0.2217 | 2.41 | | 90 | 0.2321 | 0.2321 | |

According to the theory the resistance should vary as the cube of the sine, whereas from an angle of 90° it decreases in a less ratio, but not as any constant power, nor as any function of the sine and cosine. Hence the actual resistance always exceeds that which is deduced from theory, assuming the perpendicular resistance to be the same. The cause of this difference is partly owing to our theory neglecting that part of the force which after resolution acts parallel to the plane, but which, according to experiments, is really a part of the force which acts upon the plane.

**Table V. Containing the Resistance of a Plane struck by the Fluid in Motion, and inclined at different angles to the direction of its path.**

| Angle of inclination | Resistance by experiment | Resistance by theory | |----------------------|--------------------------|---------------------| | Degrees | Oz. dwts. grs. | Oz. dwts. grs. | | 90 | 1 17 12 | 1 17 12 | | 80 | 1 17 0 | 1 16 22 | | 70 | 1 15 12 | 1 15 6 | | 60 | 1 12 12 | 1 12 11 | | 50 | 1 18 10 | 1 18 17 | | 40 | 1 4 10 | 1 4 2 | | 30 | 0 18 18 | 0 18 18 | | 20 | 0 12 12 | 0 12 19 | | 10 | 0 6 4 | 0 6 12 |

It appears from the preceding results, that the resistance varies as the sine of the angle at which the fluid strikes the plane, the difference between theory and experiment being such as might be expected from the necessary inaccuracy of the experiments.

By comparing the preceding table with Table IV., it will be found that the resistance of a plane moving in a fluid is to the resistance of the same plane when struck by the fluid in motion as 5 to 6. In both these cases the actual effect on the plane must be the same, and therefore, the difference in the resistance can arise only from the action of the fluid behind the body in the former case.

**CHAPTER IV. ON THE OSCILLATION OF FLUIDS, AND THE UNDULATION OF WAVES.**

**Prop. I.**

The oscillations of water in a syphon, consisting of two vertical branches and a horizontal one, are isochronous, and have the same duration as the oscillations of a pendulum, whose length is equal to half the length of the oscillating column of water.

Into the tube MNOP, having its internal diameter everywhere the same, introduce a quantity of water. When the water is in equilibrium, the two surfaces AB, CD will be in the same horizontal line AD. If this equilibrium be disturbed by making the syphon oscillate round the point y, the water will rise and fall alternately in the vertical branches after the syphon is at rest. Suppose the water to rise to EF in the branch MO, it will evidently fall to GH in the other branch, so that CG is equal to AE. Then it is evident, that the force which makes the water oscillate, is the weight of the column EFKL, which is double the column EABF; and that this force is to the whole weight of the water, as 2 AE is to AOPD. Now, let P be a pendulum, whose length is equal to half the length of the oscillating column AOPD, and which describes to the lowest point Sarches PS, equal to AE; then 2 AE : AOPD = AE : QP, because AE is one-half of 2 AE, and QP one-half of AOPD. Consequently, since AOPD is a constant quantity, the force which makes the water oscillate is always proportional to the space which it runs through, and its oscillations are therefore isochronous. The force Oscillation which makes the pendulum of Fluids describe the arch PS, is to the weight of the pendulum as PS is to PQ, or as AE is to PQ; since AE = PS; but the force which makes the water oscillate, is to the weight of the whole water in the same ratio; consequently, since the pendulum P, and the column AOPD, are influenced by the very same force, their oscillations must be performed in the same time. Q.E.D.

296. Cor. As the oscillations of water and of pendulums are regulated by the same laws, if the oscillating column of water is increased or diminished, the time in which the oscillations are performed will increase or diminish in the subduplicate ratio of the length of the pendulum.

Scholium.

297. This subject has been treated in a general manner, by Newton and different philosophers, who have shewn how to determine the time of an oscillation, whatever be the form of the syphon. See the Principia, lib. ii. Prop. 45, 46. Bossut's Traité d'Hydrodynamique, tom. i. Notes sur le Chap. II. Part II. Bernoulli Opera, tom. iii. p. 125, and Encyclopédie, art. Ondes.

Prop. II.

On the undulation of waves.

298. The undulations of waves are performed in the same time as the oscillations of a pendulum whose length is equal to the breadth of a wave, or to the distance between two neighbouring cavities or eminences.

In the waves ABCDEF, the undulations are performed in such a manner, that the highest parts ACE become the lowest; and as the force which depresses the eminences ACE is always the weight of water contained in these eminences, it is obvious, that the undulations of waves are of the same kind as the undulations or oscillations of water in a syphon. It follows, therefore, from Prop. I. that if we take a pendulum, whose length is one-half BM, or half the distance between the highest and lowest parts of the wave, the highest parts of each wave will descend to the lowest of its parts during one oscillation of the pendulum, and in the time of another oscillation will again become the highest parts. The pendulum, therefore, will perform two oscillations in the time that each wave performs one undulation, that is, in the time that each wave describes the space AC or BD, between two neighbouring eminences or cavities, which is called the breadth of the wave. Now, if a pendulum, whose length is one-half BM, performs two oscillations in the above time, it will require a pendulum four times that length to perform only one oscillation in the same time, that is, a pendulum whose length is AC or BD, since $4 \times \frac{1}{2} BM = 2 BM = AC$ or BD. Q.E.D.

Scholium.

299. The explanation of the oscillation of waves contained in the two preceding propositions, was first given by Sir Isaac Newton, in his Principia, lib. ii. Prop. 44. He considered it only as an approximation to the truth, since it supposes the waves to rise and fall perpendicularly like the water in the vertical branches of the syphon, while their real motion is partly circular. The theory of Newton was, nevertheless, adopted by succeeding philosophers, and gave rise to many analogous discussions respecting the undulation of waves. Very lately, however, an attempt has been made by M. Flaugergues, to overturn the theory of Newton. From a number of experiments on the motion and figure of waves, an account of which may be seen in the Journal des Sciences, for October 1789, M. Flaugergues concludes, that a wave is not the result of a motion in the particles of water, by which they ascend and descend alternately in a serpentine line, when moving from the place where the water received the shock; but that it is an intumescence which this shock occasions around the place where it is received, by the depression that is there produced. This intumescence afterwards propagates itself circularly, while it removes from the place where the shock first raised it above the level of the stagnant water. A portion of the stagnant water then flows from all sides into the hollow formed at the place where the shock was received; this hollow is thus heaped with fluid, and the water is elevated so as to produce all around another intumescence, or a new wave, which propagates itself circularly as before. The repetition of this effect produces on the surface of the water a number of concentric rings, successively elevated and depressed, which have the appearance of an undulatory motion. This interesting subject has also been discussed by M. La Grange, in his Mécanique Analytique, to which we must refer the reader for farther information. See History of Hydrodynamics, art. 21.

PART III.—ON HYDRAULIC MACHINERY.

300. To describe the various machines in which water is the impelling power, would be an endless and unprofitable task. Those machines which can be driven by wind, steam, and the force of men or horses, as well as they can be driven by water, do not properly belong to the science of hydraulics. By hydraulic machinery, therefore, we are to understand those various contrivances by which water can be employed as the impelling power of machinery; and those machines which are employed to raise water, or which could not operate without the assistance of that fluid.

CHAPTER I. ON WATER-WHEELS.

301. Water-wheels are divided into three kinds, Overshot-wheels, Breast-wheels, and Undershot-wheels, which derive their names from the manner in which the water is delivered upon their circumferences.

Sect. I. On Overshot-Wheels.

302. An overshot-wheel is a wheel driven by the weight of water, conveyed into buckets disposed on its circumference. The canal MN conveys the water into the second bucket from the top Aa. The equilibrium of the wheel is therefore destroyed; and the power of the bucket Aa, to turn the wheel round its centre of motion O, is the same as if the weight of the water in the bucket were suspended at m, the extremity of the lever Om, c being the centre of gravity of the bucket, and Om a perpendicular let fall from the fulcrum O to the direction cm, in which the force is exerted. In consequence of this destruction of equilibrium, the wheel will move round in the direction AB, the bucket Aa will be at d, and the empty bucket b will take the place of Aa, and receive water from the spout N. The force acting on the wheel is now the water in the bucket d acting with a lever nO, and the water in the bucket Aa acting with a lever mO. The velocity of the wheel will therefore increase with the number of loaded buckets, and with their distance from the vertex of the wheel; for the lever by which they tend to turn the wheel about its axis, increases as the buckets approach to c, where their power, represented by eO, is a maximum. After the buckets have passed c, the lever by which they act gradually diminishes, they lose by degrees a small portion of their water; and as soon as they reach B it is completely discharged. When the wheel begins to move, its velocity will increase rapidly till the quadrant of buckets bc is completely filled. While these buckets are descending through the inferior quadrant eP, and the buckets on the left hand of b are receiving water from the spout, the velocity of the wheel will still increase; but the increments of velocity will be smaller and smaller, since the levers by which the inferior buckets act are gradually diminishing. As soon as the highest bucket Ac has reached the point B, where it is emptied, the whole semicircumference nearly of the wheel is loaded with water; and when the bucket at B is discharging its contents, the bucket at A is filling, so that the load in the buckets, by which the wheel is impelled, will be always the same, and the velocity of the wheel will become uniform.

303. In order to find the power of the loaded arch to turn the wheel, or, which is the same thing, to find a weight which, suspended at the opposite extremity C, will balance the loaded arch or keep it in equilibrium, we must multiply the weight of water in each bucket by the length of the virtual lever by which it acts, and take the sum of all these momenta for the momentum of the loaded arch. It will be much easier, however, and the result will be the same, if we multiply the weight of all the water on the arch AB, on Water-Wheels, by the distance of its centre of gravity G, from the fulcrum or centre of motion O. Now, by the property of the centre of gravity (see Mechanics), the distance of the centre of gravity of a circular arch from its centre, is a fourth proportional to half the arch, the radius, and the sine of half the arch. Since the vertical bucket b has no power to turn the wheel if it were filled, and since two or three buckets between B and P are always empty, we may safely suppose that the loaded arch never exceeds 160°, so that if R = radius of the wheel in feet, we shall have the length of half the loaded arch, or $80^\circ = 2R \times 3.1416 \times \frac{80}{360} = R \times 1.396$; and the distance of the centre of gravity from the fulcrum O, $GO = \frac{R \times \sin 80^\circ}{R \times 1.396}$. Now, if N be the number of buckets in the wheel, $\frac{160N}{360}$ or $\frac{4N}{9}$, will be the number of buckets in the loaded arch; and if G be the number of ale gallons contained in each bucket, the weight of the water in each bucket will be $1.02 \times G$, pounds avoirdupois. The weight of the water, therefore, in the loaded arch, will be $\frac{4N}{9} \times 102G$, and consequently the momentum of the loaded arch will be

$$\frac{4N}{9} \times 10.2G \times \frac{R \times \sin 80^\circ}{R \times 1.396} = \frac{4N}{9} \times 10.2G \times 0.6338 = \frac{4N}{9} \times 6.465G$$ pounds avoirdupois. Hence, we have the following rule: Multiply the constant number 6.465 by $\frac{4}{9}$ of the number of buckets in the wheel, and this product by the number of ale gallons in each bucket; and the result will be the effective weight, or momentum of the water in the loaded arch. For a description of the best form that can be given to the buckets, see the article WATER-WORKS. Dr Robison has there recommended a mode of constructing the buckets invented by Mr Burns, who divided each bucket into two by means of a partition; but the writer of this article is assured, on the authority of an ingenious mill-wright, who wrought with Mr Burns at the time when wheels of this kind were constructed, that the inner bucket is never filled with water, and that much of the power is thus lost. The partition prevents the introduction of the fluid, and the water is driven backwards by the escape of the included air.

304. In order to determine the best form of the buckets, we must consider that the power of the wheel would be a maximum, if the whole of its semi-circumference were loaded with water. This effect would be obtained if the buckets had the shape shown in fig. 81., where ABC is the form of the bucket, AB being a continuation of the radius, and BC part of the circumference of the wheel, and nearly equal to AD. But as a small aperture at CE will neither admit nor discharge the water, the form shown in fig. 82. has been proposed by Sir David Brewster as the best. In this construction, BC is made a little larger than BE, and AB is diminished so as to make the angle ABC a little greater than 90°. The angles AB should be rounded off, so as to make ABC a curve, as indicated by the dotted line. The aperture at dE must be sufficient for the introduction and discharge of the water, and the side BE of the bucket should be as smooth and even as possible.

305. The construction of an overshot-wheel, and the mode of admitting the water into the buckets from the mill-course MN, is shewn in the following figure. In an overshot- On Water-wheel constructed by Mr Smeaton, the wheel is exactly the height of the fall, and in another it exceeds that height, so as to be intermediate between an overshot and a breast wheel.

306. In the construction of overshot-wheels, it is of great importance to determine what should be the diameter of the wheel relatively to the height of the fall. It is evident that its diameter cannot exceed the height of the fall. Some mechanical writers have demonstrated that, in theory, an overshot-wheel will produce a maximum effect when its diameter is two-thirds of that height, the water being supposed to fall into the buckets with the velocity of the wheel. But this rule is palpably erroneous, and directly repugnant to the results of experiment. For if the height of the fall be 48 feet, the diameter of the wheel will, according to this rule, be 32 feet; and the water having to fall through 16 feet before it reaches the buckets, will have a velocity of 32 feet per second, which, according to the hypothesis, must also be the velocity of the wheel's circumference. But Smeaton has proved, that a maximum effect is produced by an overshot-wheel of any diameter, when its velocity is only three feet per second. The Chevalier de Borda has shewn, that overshot-wheels will produce a maximum effect when their diameter is equal to the height of the fall; and this is completely confirmed by Mr Smeaton's experiments. From a great number of trials, Mr Smeaton has concluded, "that the higher the wheel is in proportion to the whole descent, the greater will be the effect." Nor is it difficult to assign the reason of this. The water which is conveyed into the buckets can produce very little effect by its impulse, even if its velocity be great; both on account of the obliquity with which it strikes the buckets, and in consequence of the loss of water occasioned by a considerable quantity of the fluid being dashed over their sides. Instead, therefore, of expecting an increase of effect from the impulse of the water occasioned by its fall through one-third of the whole height, we should allow it to act through this height by its gravity, and therefore make the diameter of the wheel as great as possible. But a disadvantage attends even this rule; for if the water is conveyed into the buckets without any velocity, which must be the case when the diameter of the wheel equals the height of the fall, the velocity of the wheel will be retarded by the impulse of the buckets against the water, and much power would be lost by the water dashing over them. In order, therefore, to avoid all inconveniences, the distance of the spout from the receiving bucket should, in general, be about two or three inches, that the water may be delivered with a velocity a little greater than that of the wheel; or, in other words, the diameter of an overshot-wheel should be two or three inches less than the greatest height of the fall; and yet it is no uncommon thing to see the diameters of these wheels scarcely one-half of that height.

307. The proper velocity of overshot-wheels is a subject on which mechanical writers have entertained different sentiments. While some have maintained that there is a certain velocity which produces a maximum effect, Deparcieux has endeavoured to prove, by a set of ingenious experiments, that most work is performed by an overshot-wheel when it moves slowly, and that the more its motion is retarded by increasing the work to be performed, the greater will be the performance of the wheel. In these experiments he employed a small wheel, 20 inches in diameter, having its circumference furnished with 48 buckets, evenly distributed. On the centre or axle of this wheel were placed 4 cylinders of different diameters, the first being 1 inch in diameter, the second 2 inches, the third 3 inches, and the fourth 4 inches. When the experiments are made, a cord is attached to one of the cylinders, and after passing over a pulley, a weight is suspended at its other extremity. By moving the wheel upon its axis, the cord winds round the cylinder and raises the weight. In order to diminish the friction, the gudgeons of the wheel are supported by two friction rollers, and before the wheel, a little higher than its axis, is placed a small table which supports a vessel filled with water, having an orifice in the side next the wheel. Above this vessel is placed a large bottle full of water and inverted, having its mouth immersed a few lines in the water, so that it empties itself in proportion as the water in the vessel is discharged from the orifice. The quantity of water thus discharged is always the same, and is conveyed from the orifice by means of a canal to the buckets of the wheel. With this apparatus he obtained the following results:

| Diameters of the Cylinders | Altitude through which 12 ounces were elevated | Altitude through which 24 ounces were elevated | |-----------------------------|-----------------------------------------------|-----------------------------------------------| | Inches | Inches Lines | Inches Lines | | 1 | 60 9 | 40 0 | | 2 | 80 6 | 43 6 | | 3 | 85 6 | 44 6 | | 4 | 87 9 | 45 3 |

308. When the large cylinders were used, the velocity of the wheel was smaller, because the resistances are proportional to their diameter, the weight being the same. Hence it appears, by comparing the four results in column 2d with one another, and also the four results in column 3d, that when the wheel turns more slowly, the effect, which is in this case measured by the elevation of the raised weight always increases. When the weight of 24 ounces was used, the resistance was twice as great, and the velocity twice as slow, as when the 12 ounce weight was employed. But by comparing the results in column 2d with the corresponding results in column 3d, it appears, that when the 24 ounce weight was employed, and the velocity was only one-half of what it was when the 12 ounce weight was used, the effect was more than one-half; the numbers in the 3d column being more than one-half the numbers in the 2d. Hence we may conclude, that the slower an overshot-wheel moves, the greater will be its performance.

309. These experiments of Deparcieux presented such slower unexpected results, as to induce other philosophers to examine them with care. The Chevalier D'Arcy, in particular, considered them attentively. He maintained that there was a determinate velocity when the effect of a wheel reached its maximum; and he has shown, by comparing the experiments of Deparcieux with his own formulae, that the overshot-wheel which Deparcieux employed never effect... moved with such a small velocity as corresponded with the maximum effect, and that if he had increased the diameter of his cylinders, or the magnitude of the weights, his own experiments would have exhibited the degree of velocity, when the effect was the greatest possible.

310. The reasoning of the Chevalier D'Arcy is completely confirmed by the experiments of Smeaton. This celebrated engineer concludes with Deparcieux, that, *ceteris paribus*, the less the velocity of the wheel, the greater will be its effect. But he observes, on the contrary, that when the wheel of his model made about 30 turns in a minute, the effect was nearly the greatest; when it made 30 turns, the effect was diminished about one-twentieth part; and that when it made 40 it was diminished about one-fourth; when it made less than 18½ turns, its motion was irregular, and when it was loaded so that it could not make 18 turns, the wheel was overpowered by its load. Mr Smeaton likewise observes, that when the circumferences of overshot-wheels, whether high or low, move with the velocity of three feet per second, and when the other parts of the work are properly adapted to it, they will produce the greatest possible effect. He allows, however, that high wheels may deviate farther from this rule before losing their power than low ones can be permitted to do; and assures us that he has seen a wheel 24 feet high moving at Wheels. the rate of six feet per second, without losing any considerable part of its power, and likewise a wheel 33 feet high moving very steadily and well with a velocity but little exceeding two feet.

311. The experiments of the Abbé Bossut may also be adduced in support of the same reasoning. He employed a wheel 3 feet in diameter, furnished with 48 buckets, having each three inches of depth, and four inches of width. The canal which conveyed the water into the buckets was perfectly horizontal, and was five inches wide. It furnished uniformly 1194 cubic inches of water in a minute. The resistance to be overcome was a variety of weights fixed to the extremity of a cord, which, after passing over a pulley as in Deparcieux's experiments, winded round the cylindrical axle of the wheel. The diameter of this cylinder was two inches and seven lines, and that of the gudgeons or pivots of the wheel two lines and a half. The number of turns which the wheel made in a minute was not reckoned till its motion became uniform, which always happened when it had performed five or six revolutions. When the wheel was unloaded it made 40½ turns in a minute.

| Number of pounds raised | Number of seconds in which the load was raised | Number of revolutions performed by the wheel | Effect of the wheel, or the product of the number of turns multiplied by the load | |------------------------|-----------------------------------------------|---------------------------------------------|--------------------------------------------------------------------------------| | 11 | 60" | 11½ | 131½ | | 12 | 60 | 11½ | 134½ | | 13 | 60 | 10½ | 136½ | | 14 | 60 | 9½ | 137½ | | 15 | 60 | 9½ | 138½ | | 16 | 60 | 8½ | 138½ | | 17 | 60 | 8½ | 139½ | | 18 | 60 | 7½ | 138 | | 19 | The wheel turned but exceedingly slow. | | | | 20 | The wheel stopped though first put in motion by the hand to make it catch the water. | | |

312. It appears evidently from the last column, which we have computed on purpose, that the effect increases as the velocity diminishes; but that the effect is a maximum when the number of turns is 8½ in a minute, being then 139½. When the velocity was farther diminished by adding an additional pound to the resistance, the effect was diminished to 138, and when the velocity was still less, the wheel ceased to move.

Now since the wheel was three feet in diameter, and 9.42 feet in circumference, the velocity of its circumference will be about one foot four inches per second, when it performs 8½ turns in a minute, or when the maximum effect is produced. With Mr Smeaton's model, the maximum effect was produced when the velocity of the wheel's circumference was two feet per second. So that the experiments both of Smeaton and Bossut concur to prove, that the power of overshot wheels increases as the velocity diminishes; but that there is a certain velocity, between one and two feet per second, when the wheel produces a maximum effect. Since when the wheel was unloaded it turned 40½ times in a minute, and performed only 8½ revolutions when its power was a maximum, the velocity of the wheel when unloaded will be to its velocity when the effect is the greatest, as five to one, nearly.

313. The Chevalier de Borda maintains that an overshot wheel will raise through the height of the fall a quantity of water equal to that by which it is driven, and Albert Euler has shewn that the effect of these wheels is very much inferior to the momentum or force which impels them. It appears, however, from Mr Smeaton's experiments, that when the work performed was a maximum, the ratio of the power to the effect was as four to three, when the height of the fall and the quantities of water expended were the least; but that it was as four to two when the heights of the fall and the quantities discharged were the greatest. By taking a mean between these ratios, we may conclude, in general, that in overshot-wheels the power is to the effect as three to one. In this case the power is supposed to be computed from the whole height of the fall; because the water must be raised to that height in order to be in a condition of producing the same effect a second time. When the power of the water is estimated only from the height of the wheel, the ratio of the power to the effect was more constant, being nearly as five to four.

314. The theory of overshot-wheels has been ably discussed by Albert Euler and Lambert. The former of these philosophers has shewn that the altitude of the wheel should be made as great as possible; that the buckets should be made as capacious as other circumstances will permit; that their form should be such as to convey the On Water, water as near the lowest point of the wheel as can be conveniently done; and that the motion of the wheel should be slow, that the buckets may be completely filled. He has likewise shewn that the effect of the wheel increases as its velocity is diminished; and that overshot-wheels should be used only when there is a sufficient height of fall. The results of Lambert's investigations are less consonant with the experiments of Smeaton. By examining the following table, which contains these results, it will appear at once that he makes the diameter of the wheel much smaller than it ought to be.

### Table for Overshot-Mills

| Height of the fall reckoning from the surface of the stream | Radius of the wheel reckoning from the extremity of the buckets | Width of the buckets | Depth of the buckets | Velocity of the wheel per second | Time in which the wheel performs one revolution | Turns of the millstone for one of the wheel | Force of the water upon the buckets | The length of m n in Fig. 80. | The length of n o in Fig. 80. | Quantity of water required per second to turn the wheel | |-------------------------------------------------------------|---------------------------------------------------------------|---------------------|----------------------|---------------------------------|------------------------------------------|-----------------------------------|-------------------------------|--------------------------|--------------------------|----------------------------------| | Feet. | Feet. | Feet. | Feet. | Feet. | Seconds. | Lb. avoirdupois | Feet. | Feet. | Cub. Feet. | | 7 | 2.83 | 1.09 | 2.02 | 5.27 | 3.38 | 8.45 | 636 | 0.33 | 1.15 | 10.35 | | 8 | 3.22 | 1.14 | 1.44 | 5.63 | 3.61 | 9.02 | 595 | 0.38 | 1.32 | 9.23 | | 9 | 3.63 | 1.27 | 1.07 | 5.94 | 3.83 | 9.57 | 565 | 0.42 | 1.48 | 8.21 | | 10 | 4.04 | 0.43 | 0.82 | 6.30 | 4.04 | 10.10 | 531 | 0.48 | 1.65 | 7.38 | | 11 | 4.45 | 0.57 | 0.65 | 6.60 | 4.23 | 10.57 | 511 | 0.52 | 1.81 | 6.71 | | 12 | 4.86 | 0.71 | 0.52 | 6.89 | 4.42 | 11.05 | 486 | 0.57 | 1.98 | 6.15 |

### Sect. II. On Breast-Wheels.

315. A breast-wheel partakes of the nature both of an overshot and an undershot wheel, and is driven partly by the impulse, but chiefly by the weight, of the water. The mill-course (fig. 84), is made concentric with the wheel, which is fitted to it in such a manner that no water is allowed to escape at the sides and extremities of the floatboards. The water is delivered into the openings of the wheel through an iron grating a b, and its admission is regulated by two shutters c, d, the lowest of which is adjusted till a sufficient quantity of water passes over it; and the other shutter c is made to descend by machinery when the wheel is to be stopped, and the water retained in the reservoir R. According to Mr Smeaton, the effect of a wheel driven in this manner is equal "to the effect of an undershot-wheel whose head of water is equal to the difference of level between the surface of water in the reservoir, and the point where it strikes the wheel, added to that of an overshot whose height is equal to the difference of level between the point where it strikes the wheel and the level of the tail water."

316. Mr Lambert, of the Academy of Sciences at Berlin, observes, that a breast-wheel should be used when the fall of water is above four feet in height, and below ten. The following table is calculated from Lambert's formulae, and exhibits at one view the results of his investigations.

### Table for Breast-Mills

| Height of the fall in feet. | Breadth of the floatboards. | Depth of the floatboards. | Radius of the water-wheel reckoned from the extremity of the floatboards. | Velocity of the wheel per second | Time in which the wheel performs one revolution | Turns of the millstone for one of the wheel | Force of the water upon the floatboards | The length of m n in fig. 84. | The length of n o in fig. 84. | Water required per second to turn the wheel | |-----------------------------|-----------------------------|---------------------------|---------------------------------------------------------------|---------------------------------|------------------------------------------|-----------------------------------|-------------------------------|--------------------------|--------------------------|----------------------------------| | Feet. | Feet. | Feet. | Feet. | Feet. | Seconds. | Lb. avoirdupois | Feet. | Feet. | Cub. Feet. | | 1 | 0.17 | 198.6 | 0.75 | 2.18 | 1.92 | 4.80 | 1536 | 0.08 | 0.23 | 74.30 | | 2 | 0.34 | 35.1 | 1.50 | 3.09 | 2.72 | 6.80 | 1084 | 0.15 | 0.46 | 37.15 | | 3 | 0.51 | 12.7 | 2.26 | 3.78 | 3.33 | 8.32 | 886 | 0.23 | 0.68 | 24.77 | | 4 | 0.69 | 6.2 | 3.01 | 4.36 | 3.84 | 9.60 | 768 | 0.30 | 0.91 | 18.57 | | 5 | 0.86 | 3.57 | 3.76 | 4.88 | 4.28 | 10.70 | 686 | 0.38 | 1.14 | 14.86 | | 6 | 1.03 | 2.25 | 4.51 | 5.35 | 4.70 | 11.76 | 626 | 0.46 | 1.37 | 12.38 | | 7 | 1.20 | 1.53 | 5.26 | 5.77 | 5.08 | 12.70 | 581 | 0.53 | 1.60 | 10.61 | | 8 | 1.37 | 1.10 | 6.02 | 6.17 | 5.43 | 13.58 | 543 | 0.60 | 1.83 | 9.29 | | 9 | 1.54 | 0.81 | 6.77 | 6.55 | 5.76 | 14.40 | 512 | 0.68 | 2.05 | 8.26 | | 10 | 1.71 | 0.77 | 7.52 | 6.90 | 6.07 | 15.18 | 486 | 0.76 | 2.28 | 7.43 |

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* Smeaton on Mills, schol. p. 36. * Nouv. Mém. de l'Académie de Berlin. 1775, p. 71. 317. It appears from the preceding table, that when the altitude of the fall of water is below three feet, there is such an unsuitable proportion between the depth and width of the floatboards, that a breast wheel cannot well be employed. It is also evident, on the other hand, that when the height of the fall approaches to ten feet, the depth of the floatboards is too small in relation to their width. These two extremes, therefore, ought to be avoided in practice. The eleventh column of the table contains the quantity of water necessary to drive the wheel; but the total quantity of water should always exceed this, by the quantity, at least, that escapes between the mill-course and the sides and extremities of the floatboards.

Sect. III. On Undershot-Wheels.

318. An undershot-wheel is a wheel with a number of floatboards disposed on its circumference, which receive the impulse of the water conveyed to the lowest point of the wheel by an inclined canal. It is represented in fig. 85, where WW is the water-wheel, and ABDFHKMV the canal or mill-course, which conveys the water to K, where it strikes the plane floatboards no, &c., and makes the wheel revolve about its axis.

319. In order to construct the mill-course to the greatest advantage, we must give but a very small declivity to the canal which conducts the water from the river. It will be sufficient to make AB slope about one inch in 200 yards, making the declivity, however, about half an inch for the first 48 yards, in order that the water may have sufficient velocity to prevent it from falling back into the river. The inclination of the fall, represented by the angle GCR, should be 25° 50', or CR, the radius, should be to GR, the tangent of this angle, as 100 to 28, or as 25 to 12; and since the surface of the water S6 is bent from ab into ac before it is precipitated down the fall, it will be necessary to incavate the upper part BCD of the course into BD, that the water in the bottom may move parallel to the water at the surface of the stream. For this purpose take the points B, D about 12 inches distance from C, and raise the perpendiculars BE, DE. The point of intersection E will be the centre from which the arch BD is to be described; the radius being about 10 inches. Now, in order that the water may act more advantageously upon the floatboards of the wheel WW, it must assume a horizontal direction, with the same velocity which it would have acquired when it came to the point G. But, if the water were allowed to fall from C to G, it would dash upon the horizontal part HG, and thus lose a great part of its velocity. It will be necessary, therefore, to make it move along FH, an arch of a circle to which DF and KH are tangents in the points F and H. For this purpose make GF and GH each equal to three feet; and raise the perpendiculars HI, FI which will intersect one another in the point I, distant about four feet nine inches from the points F and H, and the centre of the arch FH will be determined. The distance HK, through which the water runs before it acts upon the wheel, should not be less than two or three feet, in order that the different filaments of the fluid may have attained a horizontal direction. If HK were too large, the stream would suffer a diminution of velocity by its friction on the bottom of the course. That no water may escape between the bottom of the course KH and the extremities of the floatboards, KL should be about three inches, and the extremity o of the floatboard no ought to reach below the line HKX, sufficient room being left between o and M for the play of the wheel; or KLM may be formed into the arch of a circle KM concentric with the wheel. The line LMV, which has been called the course of impulsion, should be prolonged so as to support the water as long as it can act upon the floatboards, and should be about nine inches distant from OP, a horizontal line passing through O the lowest point of the fall; for if OL were much less than nine inches, the water having spent the greatest part of its force in impelling the floatboard, would accumulate below the wheel, and retard its motion. For the same reason another course, which has been called the course of discharge, should be connected with LMV by the curve VN, to preserve the remaining velocity of the water, which would otherwise be discharged by falling perpendicularly from V to N. The course of discharge, which is represented by the line VZ, sloping from the point O, should be about 16 yards long, having an inch of declivity for every two yards. The canal which reconducts the water from the course of discharge to the river should slope about four inches in the first 200 yards, three inches in the second 200 yards, decreasing gradually till it terminates in the river. But if the river to which the water is conveyed should, when swelled by the rains, force the water back upon the wheel, the canal must have a greater declivity to prevent this from taking place. Hence it is evident that very accurate levelling is requisite to the proper formation of the mill-course.

As it is of great importance that none of the water should escape, either below the floatboards or at their sides, without contributing to turn the wheel, the course of impulsion KV should be wider than the course at K, as represented in fig. 86, where CD the course of impulsion corresponds with LV in fig. 85, AB corresponds with HK, and BC with KL. The breadth of the floatboards, therefore, should be wider than mn, and their extremities should reach a little below B, like no in fig. 85. When these precautions are properly taken, no water can escape without exerting its force upon the floatboards.

320. It has been disputed among philosophers, whether the wheel should be furnished with a small or a great number of floatboards. M. Pitot has shewn, that when the floatboards have different degrees of obliquity, the force of impulsion upon the different surfaces will be reciprocally as their breadths; Thus in fig. 87, the force of impulsion upon he will be to the force upon DO, as DO to he. Hence he concludes that the distance between the floatboards should be equal to one-half of the immersed arch, or that when one floatboard is at the bottom of the wheel, and perpendi-

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1 See Appendix to Ferguson's Lectures, vol. ii. p. 189, 2d edit. 2 Mémoires de l'Académie Paris, 1729, 8vo, p. 359. should be just leaving the stream, and the succeeding one FG just immersing into it. For when the three floatboards FG, DE, BC have the same position as in the figure, the whole force of the current NM will act upon DE when it is in the most advantageous position for receiving it; whereas, if another floatboard de were inserted between FG and DE, the part ig would cover DO, and by thus substituting an oblique for a perpendicular surface, the effect would be diminished in the proportion of DO to ig. Hence it is evident, that, upon this principle, the depth of the floatboard DE should be always equal to the versed sine of the arch EG.

321. Notwithstanding the plausibility of this reasoning, it will not be difficult to show that it is destitute of foundation. It is evident from fig 87, that when one of the floatboards DE is perpendicular to the stream, it receives the whole impulse of the water in the most advantageous manner. But when it arrives at the position de, and the succeeding one FG at the position fg, so that the angle eAg may be bisected by the perpendicular AE, the situation of these floatboards will be the most disadvantageous, for a great part of the water will escape between the extremities g and e of the floatboards without striking them, and the part ig of the floatboards, which is really impelled, is less than DE, and oblique to the current. The wheel, therefore, must move irregularly, sometimes quick and sometimes slow, according to the position of the floatboards with respect to the stream; and this inequality will increase with the arch plunged in the water. The reasoning of M. Pitot, indeed, is founded on the supposition, that if another floatboard fg were placed between FG and DF, it would annihilate the force of the water that impels it, and prevent any of the fluid from striking the corresponding part DO of the preceding floatboard. But this is not the case. For when the water has acted upon fg, it still retains a part of its motion, and after bending round the extremity g, strikes DE with its remaining force. We are entitled, therefore, to conclude that advantage must be gained by using more floatboards than are recommended by Pitot.

322. It is evident from the preceding remarks, that in order to remove any inequality of motion in the wheel, and prevent the water from escaping below the extremities of the floatboards, the wheel should be furnished with the greatest possible number of floatboards, without loading it too much, or enfeebling the rim on which they are fixed. This rule was first given by M. Dupetit Vaudin; and it is easily perceived, that if the mill-wright should err in using too many floatboards, this error in excess will be perfectly trifling, and that a much greater loss of power would be occasioned by an error in defect.

323. The section of the floatboards ought not to be rectangular, like abne in fig. 87, but should be bevelled like the form abmc. For if they were rectangular, the extremity b of boards would interrupt a portion of the water which would otherwise fall on the corresponding part of the preceding floatboard. In order to find the angle abm, subtract from 180 degrees the number of degrees contained in the immersed arch CEG, and the half of the remainder will be the angle required.

324. It has been maintained by M. Pitot and other philosophers, that the floatboards should be a continuation of the radius, or perpendicular to the rim, as in fig. 85. This, indeed, is true in theory, but it appears from the most unquestionable experiments, that they should be inclined to the radius. This important fact was discovered by Deparcieux in 1753, and proved by several experiments. When the floatboards are inclined, the water heaps up on their surface, and acts not only by its impulse, but also by its weight. The same truth has also been confirmed by the Abbé Bossut, the most accurate of whose experiments are contained in the following table. The wheel that was employed was immersed four inches vertically in the water, and it was furnished with 12 floatboards.

| Inclination of the floatboard | Number of pounds raised | Time in which the load was raised in seconds | Number of turns made by the wheel | |-------------------------------|------------------------|---------------------------------------------|---------------------------------| | 0 | 40 | 40 | 13\(\frac{1}{4}\) | | 15 | 40 | 40 | 14\(\frac{1}{4}\) | | 30 | 40 | 40 | 14\(\frac{1}{4}\) | | 37 | 40 | 40 | 14\(\frac{1}{4}\) |

325. It is obvious, from the preceding table, that the wheel made the greatest number of turns, or moved with the greatest velocity, when the number of floatboards was between 15 and 30. When the water-wheels are placed on canals that have little declivity, and in which the water can escape freely after its impulse upon the floatboards, it would be proper to make the floatboards a continuation of the radius. But when they move in an inclined mill-course, an augmentation of velocity may be expected from an inclination of the floatboards.

326. Having thus pointed out the most scientific method of constructing the wheel, and delivering the water upon proper floatboards, we have now to determine the velocity with which it should move. It is evident that the velocity of the wheel must be always less than that of the water which impels it, even when there is no work to be performed; for a part of the impelling power is necessarily spent in overcoming the inertia of the wheel, and the resistance of friction. It is likewise obvious, that when the wheel has little or no velocity, its performance will be very trifling. There is, consequently, a certain proportion between the velocity of the water and the wheel, when its effect is a maximum. By the reasoning which is employed in the section on undershot-wheels in the article Water-Works, Parent and Pitot found, that a maximum effect was produced when the velocity of the wheel was one-third of the velocity of the water; and Desaguliers, Maclaurin, Lambert, Atwood, have adopted their conclusions.

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1 A table containing the number of floatboards for wheels of different diameters, and founded on this principle, has been given in the Appendix to Ferguson's Lectures, vol. ii. p. 149, 2d edit. 2 Memoires des Sciences Etrangères, tom. I. 3 Desaguliers' Experimental Philosophy, vol. ii. p. 424, lect. 12. 4 Maclaurin's Fluxions, art. 907, p. 728. 5 Atwood on Rectilineal and Rotatory Motion, p. 275-284. 6 Nouv. Memoires de l'Acad. Berlin, 1775, p. 63. In the calculus from which this result was deduced, it was taken for granted that the momentum or force of water upon the wheel is in the duplicate ratio of the relative velocity, or as the square of the difference between the velocity of the water and that of the wheel. This supposition, indeed, is perfectly correct when the water impels a single floatboard; for as the number of particles which strike the floatboard in a given time, and also the momentum of these, are each as the relative velocity of the floatboards, the momentum must be as the square of the relative velocity, that is, \( M = R^2 \), \( M \) being the momentum, and \( R \) the relative velocity. But we have seen, in some of the preceding paragraphs, that the water acts on more than one floatboard at a time. Now, the number of floatboards acted upon in a given time will be as the velocity of the wheel, or inversely as the relative velocity; for if you increase the relative velocity, the velocity of the water remaining the same, you must diminish the velocity of the wheel.

Consequently, we shall have \( M = \frac{R^2}{R} \) or \( M = R \); that is, the momentum of the water acting upon the wheel is directly as the relative velocity.

327. Let \( V \) be now the velocity of the stream, and \( F \) the force with which it would strike the floatboard at rest, and \( v \) the velocity of the wheel. Then the relative velocity will be \( V - v \); and since the velocity of the water will be to its momentum, or the force with which it would strike the floatboard at rest, as the relative velocity is to the real force which the water exerts against the moving floatboards, we shall have \( V : V - v = F : F \times \frac{V - v}{V} = \frac{F}{V} \times V - v \).

But the effect of the wheel is measured by the product of the momentum of the water and the velocity of the wheel, consequently the effect of the undershot-wheel will be

\[ e \times \frac{F}{V} \times V - v = \frac{F}{V} \times V v - v^2. \]

Now, this effect is to be a maximum, and therefore its fluxion must be equal to 0, that is, \( e \) being the variable quantity, \( V v - 2 vv = 0 \), or

\[ 2 vv = V v. \]

Dividing by \( v \), we have \( 2 v = V \), and \( v = \frac{V}{2} \),

that is, the velocity of the wheel will be one-half the velocity of the fluid when the effect is a maximum.

328. This result, which was first obtained by the Chevalier de Borda, has been amply confirmed by the experiments of Mr Smeaton. "The velocity of the stream," says he, "varies at the maximum between one-third and one-half that of the water; but in all the cases in which most work is performed in proportion to the water expended, and which approach the nearest to the circumstances of great works, when properly executed, the maximum lies much nearer one-half than one-third, one-half seeming to be the true maximum, if nothing were lost by the resistance of the air, the scattering of the water carried up by the wheel," &c.

329. A result, nearly similar to this, was deduced from the experiments of Bossut. He employed a wheel whose diameter was three feet. The number of floatboards was at one time 48, and at another 24, their width being five inches, and their depth six. The experiments with the wheel, when it had 48 floatboards, were made in an inclined canal, supplied from a reservoir by an orifice two inches deep, the velocity being 300 feet in 27 seconds. The experiments with the wheel, when it had 24 floatboards, were made in a canal, contained between two vertical walls, 12 or 13 feet distant. The depth of the water was about seven or eight inches, and its mean velocity about 2740 inches in 40 seconds. The floatboards of the wheel were immersed about four inches in the stream.

330. As the effect of the machine is measured by the product of the load raised, and the time employed, it will appear, by multiplying the second and third columns, that the effect was a maximum when the load was 34 pounds, the wheel performing 20 revolutions in 40 seconds. By comparing the velocity of the centre of impression computed from the diameter of the wheel, and the number of turns which it makes in 40 seconds, with the velocity of the current, it will be found, that the velocity of the wheel, when its effect is the greatest possible, is nearly two-fifths that of the stream. From the last two columns of the table, where the effect is a maximum when the load is 60 pounds, the same conclusion may be deduced.

331. The proper velocity of the wheel being thus established, we shall proceed to point out the method of constructing a millwright's table for undershot-wheels, taking it for granted that the velocity of the wheel should be one-half the velocity of the stream, and that water moves with the same velocity as falling bodies.

1. Find the perpendicular height of the fall of water above the bottom of the mill-course, and having diminished this number by one-half the depth of the water at K, call that the height of the fall.

2. Since bodies acquire a velocity of 32.174 feet, by falling through the height of 16.087 feet; and as the velocities of falling bodies are as the square roots of the heights through which they fall, the square root of 16.087 will be to the square root of the height of the fall as 32.174 to a fourth number, which will be the velocity of the water. Therefore the velocity of the water may be always found by multiplying 32.174 by the square root of the height of the fall, and dividing that product by the square root of 16.087. Or it may be found more easily by multiplying the height of the fall by the constant quantity 64.348 = 2 × 32.174, and extracting the square root of the product. This root, abstracting from the effects of friction, will be the velocity of the water required.

3. Take one-half the velocity of the water, and it will be the velocity which must be given to the floatboards, or the number of feet they must move through in a second, in order to produce a maximum effect.

4. Divide the circumference of the wheel by the velocity of its floatboards per second, and the quotient will be the number of seconds in which the wheel revolves.

5. Divide 60 by the number last found, and the quotient will be the number of turns made by the wheel in a minute. On Water.—Or the number of revolutions performed by the wheel in a minute may be found, by multiplying the velocity of the floatboards by 60, and dividing the product by the circumference of the wheel.

6. Divide 90, the number of revolutions which a millstone, five feet diameter, should make in a minute, by the number of revolutions made by the wheel in a minute; and the quotient will be the number of turns which the millstone ought to make for one revolution of the wheel.

7. Then, as the number of revolutions of the wheel in a minute is to the number of revolutions of the millstone in a minute, so must the number of staves in the trundle be to the number of teeth in the wheel, in the nearest whole numbers that can be found.

8. Multiply the number of revolutions performed by the wheel in a minute, by the number of revolutions made by the millstone for one of the wheel, and the product will be the number of revolutions made by the millstone in a minute.

332. By these rules, the following table has been computed for a water-wheel fifteen feet in diameter, which is a good medium size, the millstone being seven feet in diameter, and revolving 90 times in a minute.

**Table I. A New Mill-Wright's Table, in which the Velocity of the Wheel is one-half the Velocity of the Stream, the effects of Friction not being considered.**

| Feet | Height of the fall of water | Velocity of the water per second, friction not being considered. | Velocity of the wheel per second, being one-half that of the water. | Revolutions of the wheel per minute, its diameter being 15 feet. | Revolutions of the millstone for one of the wheel. | Teeth in the wheel and staves in the trundle. | Revolutions of the millstone per minute by these staves and teeth. | |------|---------------------------|---------------------------------------------------------------|---------------------------------------------------------------|---------------------------------------------------------------|---------------------------------------------------------------|---------------------------------------------------------------|---------------------------------------------------------------| | 1 | 8.02 | 4.01 | 5.10 | 17.55 | 106 | 6 | 90.01 | | 2 | 11.34 | 5.67 | 7.22 | 12.47 | 87 | 7 | 90.03 | | 3 | 13.89 | 6.95 | 8.85 | 10.17 | 81 | 8 | 90.00 | | 4 | 16.04 | 8.02 | 10.20 | 8.82 | 79 | 9 | 89.96 | | 5 | 17.94 | 8.97 | 11.43 | 7.87 | 71 | 9 | 89.95 | | 6 | 19.65 | 9.82 | 12.50 | 7.20 | 65 | 9 | 90.00 | | 7 | 21.22 | 10.61 | 13.51 | 6.66 | 60 | 9 | 89.98 | | 8 | 22.60 | 11.34 | 14.45 | 6.23 | 56 | 9 | 90.02 | | 9 | 24.06 | 12.03 | 15.31 | 5.88 | 53 | 9 | 90.02 | | 10 | 25.37 | 12.69 | 16.17 | 5.57 | 56 | 10 | 90.06 | | 11 | 26.60 | 13.30 | 16.95 | 5.31 | 53 | 10 | 90.00 | | 12 | 27.79 | 13.90 | 17.70 | 5.08 | 51 | 10 | 89.91 | | 13 | 28.92 | 14.46 | 18.41 | 4.89 | 49 | 10 | 90.02 | | 14 | 30.01 | 15.01 | 19.11 | 4.71 | 47 | 10 | 90.00 | | 15 | 31.07 | 15.53 | 19.80 | 4.55 | 48 | 11 | 90.09 | | 16 | 32.09 | 16.04 | 20.40 | 4.45 | 44 | 10 | 89.96 | | 17 | 33.07 | 16.54 | 21.05 | 4.28 | 47 | 11 | 90.09 | | 18 | 34.03 | 17.02 | 21.66 | 4.16 | 50 | 12 | 90.10 | | 19 | 34.97 | 17.48 | 22.26 | 4.04 | 44 | 11 | 89.93 | | 20 | 35.97 | 17.99 | 22.86 | 3.94 | 48 | 12 | 90.07 |

333. The preceding table (Appendix to Ferguson's Lectures, vol. ii. p. 174) supposes, according to theory, that the velocity of the wheel, at the maximum effect, is one-half that of the stream, which is nearly the case in practice, when the quantities of water discharged by the stream are considerable. "When we consider, however," observes the editor of the work now quoted, "that after every precaution has been observed, a small quantity of water will escape between the mill-course and the extremities of the floatboards, and that the effect is diminished by the resistance of the air and the dispersion of water carried up by the wheel, the propriety of making the wheel move with three-sevenths the velocity of the water will appear. The Cavalier de Borda supposes it never to exceed three-eighths; and Mr Smeaton and the Abbé Bossuet found two-fifths to be the proper medium." With three-sevenths, therefore, as the best medium, which differs only \( \frac{3}{7} \) from this, the numbers in the following table have been computed. In Table I. the water was supposed to move with the same velocity as falling bodies, but owing to its friction on the mill-course, &c. this is not exactly the case. We have therefore deduced the velocity of the water in column second from the following formula,

\[ V = \sqrt{\frac{172}{3} \times Rb - \frac{Hh}{2}}, \]

in which \( V \) is the velocity of the water, \( Rb \) the absolute height of the fall, and \( Hh \) the depth of the water at the bottom of the course. The formula is founded on the experiments of Bossuet, from which it appears, that if a canal be inclined one-tenth part of its length, this additional declivity will restore that velocity to the water which was destroyed by friction."

**Table II. A New Mill-Wright's Table, in which the Velocity of the Wheel is three-sevenths of the Velocity of the Water, and the effects of Friction on the Velocity of the stream reduced to computation.**

| Feet | Height of the fall of water | Velocity of the water per second, friction being considered. | Velocity of the wheel per second, being 3.71ths that of the water. | Revolutions of the wheel per minute, its diameter being 15 feet. | Revolutions of millstone for one of the wheel. | Teeth in the wheel and staves in the trundle. | Revolutions of the millstone per minute by these staves and teeth. | |------|---------------------------|---------------------------------------------------------------|---------------------------------------------------------------|---------------------------------------------------------------|---------------------------------------------------------------|---------------------------------------------------------------|---------------------------------------------------------------| | 1 | 7.62 | 3.27 | 4.16 | 21.63 | 130 | 6 | 89.98 | | 2 | 10.77 | 4.62 | 5.88 | 15.31 | 92 | 6 | 90.02 | | 3 | 13.20 | 5.66 | 7.20 | 22.50 | 100 | 8 | 90.00 | | 4 | 15.24 | 5.53 | 8.32 | 10.81 | 97 | 9 | 89.94 | | 5 | 17.04 | 7.30 | 9.28 | 9.70 | 97 | 10 | 90.02 | | 6 | 18.67 | 8.00 | 10.19 | 8.83 | 97 | 11 | 89.98 | | 7 | 20.15 | 8.64 | 10.99 | 8.19 | 90 | 11 | 90.01 | | 8 | 21.56 | 9.24 | 11.76 | 7.65 | 84 | 11 | 89.96 | | 9 | 22.86 | 9.80 | 12.47 | 7.22 | 72 | 10 | 90.03 | | 10 | 24.10 | 10.33 | 13.15 | 6.84 | 82 | 12 | 89.95 | | 11 | 25.27 | 10.83 | 13.79 | 6.53 | 85 | 13 | 90.05 | | 12 | 26.40 | 11.31 | 14.40 | 6.25 | 72 | 12 | 90.00 | | 13 | 27.47 | 11.77 | 14.99 | 6.00 | 72 | 12 | 89.94 | | 14 | 28.51 | 12.22 | 15.56 | 5.78 | 75 | 13 | 90.04 | | 15 | 29.52 | 12.65 | 16.13 | 5.58 | 67 | 12 | 90.01 | | 16 | 30.48 | 13.06 | 16.63 | 5.41 | 65 | 12 | 89.97 | | 17 | 31.42 | 13.46 | 17.14 | 5.25 | 63 | 12 | 89.99 | | 18 | 32.33 | 13.86 | 16.65 | 5.10 | 61 | 12 | 90.01 | | 19 | 33.22 | 14.24 | 18.13 | 4.96 | 64 | 13 | 89.92 | | 20 | 34.17 | 14.64 | 18.64 | 4.83 | 58 | 12 | 89.84 |

The great hydraulic machine at Marly was found to produce a maximum effect, when its velocity was two-fifths of that of the stream. In order that the wheel may move with a velocity duly adjusted to that of the current, we would not advise the mechanic to trust to the second column of Table II. for the true velocity of the stream, or to any theoretical results, even when deduced from formulae founded on experiments. Bossut, with great justice, remarks, that "it would not be exact in practice to compute the velocity of a current from its declivity. This velocity ought to be determined by immediate experiment in every particular case." Let the velocity of the water, therefore, where it strikes the wheel, be determined by the method in the following paragraph. With this velocity, as an argument, enter column second of either of these tables, according as the velocity of the wheel is to be one-half or three-sevenths that of the stream, and take out the other numbers from the table.

Various methods have been proposed by different philosophers for measuring the velocity of running water; during the electi- the method, by floating bodies, which Mariotte employed, the bent tube of Pitot, the regulator of Guglielmini, the quadrant, the little wheel, and the method proposed by the Abbé Mann; have each their advantages and disadvantages. The little wheel was employed in the experiments of Bossut. It is the most convenient mode of determining the superficial velocity of the water; and, when constructed in the following manner, will be more accurate, it is hoped, than any instrument that has hitherto been used. The small wheel should be formed of the lightest materials. It should be about ten or twelve inches in diameter, and furnished with fourteen or sixteen floatboards. This wheel moves upon a delicate screw passing through its axle B; and when impelled by the stream it will gradually approach towards D, each revolution of the wheel corresponding with a thread of the screw. The number of revolutions performed in a given time are determined upon the scale m o, by means of the index fixed at O, and moveable with the wheel, each division of the scale being equal to the breadth of a thread of the screw, and the extremity h of the index coinciding with the beginning of the scale, when the shoulder b of the wheel is screwed close to a. The parts of a revolution are indicated by the bent index m n, pointing to the periphery of the wheel, which is divided into 100 parts. When this instrument is to be used, take it by the handles CD, or when great accuracy is required, make it rest on the handleless CD; and screw the shoulder b of the wheel close to a, so that the indices may both point to o the commencement of the scales. Then, by means of a stop-watch or pendulum, find how many revolutions of the wheel are performed in a given time. Multiply the mean circumference of the wheel (or the circumference deduced from the mean radius, which is equal to the distance of the centre of impulsion or impression from the axis b B) by the number of revolutions, and the product will be the number of feet through which the water moves in the given time. On account of the friction of the screw, the resistance of the air, and the weight of the wheel, its centre of impulsion will revolve with a little less velocity than that of the stream; but the diminution of velocity, arising from these causes, may be estimated with sufficient precision for all the purposes of the practical mechanic. (Appendix to Ferguson's Lectures, vol. ii. p. 177.)

It appears, from a comparison of the numerous and accurate experiments of Mr Smeaton, that, in undershot wheels, the power employed to turn the wheel is to the effect produced as 3 to 1; and that the load which the wheel will carry at its maximum, is to the load which will totally stop it, as 3 to 4. The same experiments inform us, that the impulse of the water on the wheel, in the case of a maximum, is more than double of what is assigned by theory, that is, instead of four-sevenths of the column, it is nearly equal to the whole column. In order to account for this, Mr Smeaton observes, that the wheel was not, in this case, placed in an open river, where the natural current, after it had communicated its impulse to the float, has room on all sides to escape, as the theory supposes; but in a conduit or race, to which the float being adapted, the water could not otherwise escape than by moving along with the wheel. He likewise remarks, that when a wheel works in this manner, the water, as soon as it meets the float, receives a sudden check, and rises up against it like a wave against a fixed object; insomuch, that when the sheet of water is not a quarter of an inch thick before it meets the float, yet this sheet will act upon the whole surface of a float, whose height is three inches. Were the float, therefore, no higher than the thickness of the sheet of water, as the theory supposes, a great part of the force would be lost by the water dashing over it. In order to try what would be the effect of diminishing the number of floatboards, Mr Smeaton reduced the floatboards, which were originally 24 to 12. This change produced a diminution of the effect, as a greater quantity of water escaped between the floats and the floor. But when a circular sweep was adapted to the floor, and made of such a length that one float entered the curve before the preceding one quitted it, the effect came so near to the former, as to afford no hopes of increasing it by augmenting the number of floats beyond 24 in this particular wheel. Mr Smeaton likewise deduced, from his experiments, the following maxims:

1. That the virtual or effective head being the same, the effect will be nearly as the quantity of water expended. 2. That the expense of water being the same, the effect will be nearly as the height of the virtual or effective head. 3. That the quantity of water expended being the same, the effect is nearly as the square of the velocity. 4. The aperture being the same, the effect will be nearly as the cube of the velocity of the water.

We have hitherto supposed the floatboards, though inclined to the radius, to be perpendicular to the plane of the wheel. Undershot-wheels, however, have sometimes been constructed with floatboards inclined to the plane of the wheel. A wheel of this kind is represented in fig. 89, where AB is the wheel, and CDEFGH the oblique floatboards. The horizontal current MN is delivered on the wheel, floatboards, so as to strike them perpendicularly. On account of the size of the floatboards, every filament of the water contributes to turn the wheel; and therefore its effect will be greater than in undershot wheels of the common form. Albert Euler imagines that the effect will be twice as great, and observes, that in order to produce such an effect, the velocity of the centre of impression should be to the velocity of the water, as radius is to triple the sine of the angle by which the floatboards are inclined to the plane of the wheel. If this inclination, therefore, be $60^\circ$, the velocity of the wheel at the centre of impression ought to be to the velocity of the impelling fluid as $1$ to $\frac{3\sqrt{3}}{2}$, that is,

$$ \text{as } 5 \text{ to } 13 \text{ nearly, because } \sin 60^\circ = \frac{\sqrt{3}}{2}. $$

When the inclination is $30^\circ$, the ratio of the velocities will be found to be as $2$ to $3$.

338. In wheels of this kind, the floats may also be advantageously inclined to the radius. In this case, the stream, which still strikes them perpendicularly, is inclined to the horizon. If the angle formed by the common section of the wheel and floatboards with the radius of the wheel, be $= m$; and if the angle by which the floatboards are inclined to the plane of the wheel be $= n$, then the angle which the floatboards should form with the direction in which the wheel moves, will be $= \cos m \times \sin n$. In order, therefore, that the stream may strike the floatboards with a perpendicular impulse, its inclination to the horizon must be $= m$, and its inclination to the plane of the wheel $= 90^\circ - n$. The less that the velocity of the water is, the greater should be the angle $m$; for there is, in this case, no danger that the celerity of the wheel be too great. The area of the floatboards ought to be much greater than the section of the current; and the interval between two adjacent floatboards should be so great, that before the one completely withdraws itself from the action of the water, the other should begin to receive its impulse.

339. Horizontal water-wheels have been much used on the Continent, and are strongly recommended to our notice by the simplicity of their construction. In fig. 90, AB is the large water-wheel which moves horizontally upon its arbor CD. This arbor passes through the immovable millstone EF at D, and being fixed to the upper one GH, carries it once round for every revolution of the great wheel. The mill-course is constructed in the same manner for horizontal as for vertical wheels, with this difference only, that the part $mBnC$, fig. 85 of which KL in fig. 86 is a section, instead of being rectilineal like $mn$, must be circular like $mp$, and concentric with the rim of the wheel, sufficient room being left between it and the tips of the floatboards for the play of the wheel. In this construction, where the water moves in a horizontal direction before it strikes the wheel, the floatboards should be inclined about $25^\circ$ to the plane of the wheel, and the same number of degrees to the radius, so that the lowest and outermost sides of the floatboards may be farthest up the stream.

340. Instead of making the canal horizontal before it delivers the water on the floatboards, they are frequently inclined in such a manner as to receive the impulse perpendicularly, and in the direction of the declivity of the mill-course. When this construction is adopted, the maximum effect will be produced when the velocity of the floatboards is not less than

$$ \frac{5.67\sqrt{H}}{2 \sin A}, $$

where $H$ represents the height of the fall, and $A$ the angle which the direction of the fall makes with a vertical line. But as the quantity

$$ \frac{5.67\sqrt{H}}{2 \sin A} $$

evidently increases as the sine of $A$ decreases, it follows, that without lessening the effect of these wheels, we may diminish the angle $A$, and thus augment considerably the velocity of the floatboards, according to the nature of the machinery employed; whereas, in vertical wheels, there is only one determinate velocity which produces a maximum effect.

341. In the southern provinces of France, where horizontal wheels are generally employed, the floatboards are usually made of a curvilinear form, so as to be concave towards the stream. The Chevalier de Borda observes, that in theory a double effect is produced when the floatboards are concave; but that effect is diminished in practice, from the difficulty of making the fluid enter and leave the curve in a proper direction. Notwithstanding this difficulty, however, and other defects which might be pointed out, horizontal wheels with concave floatboards are always superior to those in which the floatboards are plain, and even to vertical wheels, when there is a sufficient fall of water. When the floatboards are plane, the wheel is driven merely by the impulse of the stream; but when they are concave, a part of the water acts by its weight and increases the velocity of the wheel. If the fall of water be 5 or 6 feet, a horizontal wheel with concave floatboards may be erected, whose maximum effect will be to that of the ordinary vertical wheels as 3 to 2.

342. An advantage attending horizontal wheels is, that the water may be divided into several canals, and delivered upon several floatboards at the same time. Each stream will heap up on its corresponding floatboard, and produce a greater effect than if the force of the water had been concentrated on a single floatboard. Horizontal wheels may be employed with greatest advantage when a small quantity of water falls through a considerable height.

343. It has been disputed among mechanical philosophers, whether overshot or undershot wheels produce the greatest effect. M. Belidor maintained that the former were inferior to the latter, while a contrary opinion was entertained by Desaguliers. It appears, however, from Mr Smeaton's experiments, that in overshot-wheels the power is to the effect nearly as 3 to 2 or as 5 to 4 in general, whereas in undershot-wheels it is only as 3 to 1. The effect of overshot-wheels, therefore, is nearly double that of undershot-wheels, other circumstances being the same. In comparing the relative effects of water-wheels, the Chevalier de Borda remarks that overshot wheels will raise through the height of the fall, a quantity of water equal to that by which they are driven; that undershot vertical wheels will produce only three-eighths of this effect; that horizontal wheels will produce a little less than one-half of it when the floatboards are plain, and a little more than one-half of it when the floatboards have a curvilinear form. Besant's Undershot-Wheel.

344. The water-wheel invented by Mr Besant of Brompton is constructed in the form of a hollow drum, so as to resist the admission of the water. The floatboards are fixed obliquely in pairs on the periphery of the wheel, so that each pair may form an acute angle open at its vertex, while one of the floatboards extends beyond the vertex of the angle. A section of the water-wheel is represented in fig. 91, where AB is the wheel, CD its axis, and mn op the position of the floatboards. The motion of common undershot-wheels is greatly retarded by the resistance which the tail-water and the atmosphere oppose to the ascending floatboards; but in Besant's wheel this resistance is greatly diminished, as the floats emerge from the stream in an oblique direction. Although this wheel is much heavier than those of the common construction, yet it revolves more easily upon its axis, as the stream has a tendency to make it float.

Conical Horizontal Wheel with Spiral Floatboards.

345. In Guyenne and Languedoc, in the south of France, a kind of conical horizontal wheel is sometimes employed for turning machinery. It is constructed in the form of an inverted cone AB, fig. 92, with spiral floatboards winding round its surface. The wheel moves on a vertical axis AB, in the building DD, and is driven chiefly by the impulse of the water conveyed by the canal C to the oblique floatboards, the direction of the current being perpendicular to the floatboards at the place of impact. When the impulsive force of the water is annihilated, it descends along the spirals, and continues to act by its weight till it reaches the bottom, when it is carried off by the canal M.

CHAPTER II. ON MACHINES DRIVEN BY THE REACTION OF WATER.

346. We have hitherto considered the mechanical effects of water as the impelling power of machinery, when it acts either by its impulse or by its gravity. The reaction of water may be employed to communicate motion by its machinery; and though this principle has not yet been adopted in practice, it appears from theory and from some detached experiments on a small scale, that a given quantity of water, falling through a given height, will produce greater effects by its reaction than by its impulse or its weight.

Sect. I. On Dr Barker's Mill.

347. This machine, which is sometimes called Parent's mill, is represented in fig. 93; where MN is the canal that conveys the water into the upright tube TT, which communicates with the horizontal arm AB. The water will therefore descend through the upright tube into this arm, and will exert upon the inside of it a pressure proportioned to the height of the fall. But if two orifices A and B be perforated at the extremities of the arm, and on contrary sides, the pressure upon these orifices will be removed by the efflux of the water, and the unbalanced pressure upon the opposite sides of the arm will make the tube and the horizontal arm revolve upon the spindle D as an axis. This will be more easily understood, if we suppose the orifices to be shut up, and consider the pressure upon a circular inch of the arm opposite to the orifice, the orifice being of the same size. The pressure upon this circular inch will be equal to a cylinder of water whose base is one inch in diameter, and whose altitude is the height of the fall; and the same force is exerted upon the shut-up orifice. These two pressures, therefore, being equal and opposite, the arm A will remain at rest. But as soon as you open the orifice, the water will issue with a velocity due to the height of the fall; the pressure upon the orifice will of consequence be removed; and as the pressure upon the circular inch opposite to the orifice still con- Machines, the equilibrium will be destroyed, the arm will move driven by in a retrograde direction.

348. The upright spindle D, on which the arm revolves, is fixed in the bottom of the arm, and screwed to it below by a nut. It is fixed to the upright tube by two cross bars, so as to move along with it. If a corn-mill is to be driven, the top of the spindle is fixed into the upper millstone m. The lower quiescent millstone n rests upon the floor K, in which is a hole, to let the meal pass into a trough below. The bridgertree GF, which supports the millstone, tube, &c. is moveable on a pin at h, and its other end is supported by an iron rod fixed into it, the top of the rod going through the fixed bracket O, furnished with a nut o. By screwing this nut, the millstone may be raised or lowered at pleasure. If any other kind of machinery is to be driven, the spindle D must be prolonged above the hopper H, and a small wheel fixed to its extremity, which will communicate its motion to any species of mechanism.

An improvement on this machine by M. Mathon de la Cour, and some excellent observations on the subject by Professor Robison, will be found in the article WATER-WORKS.

349. Mr Waring of the American Philosophical Society, has given a theory of Barker's mill with the improvement of M. Mathon de la Cour, which he has strangely ascribed to a Mr Rumsey about twenty years after it was published in Rozier's Journal de Physique, Jan. and August 1775. Contrary to every other philosopher, he makes the effect of the machine equal only to that of a good undershot-wheel, moved with the same quantity of water, falling through the same height. The following rules, however, deduced from his calculus, may be of use to those who may wish to make experiments on the effect of this interesting machine.

1. Make the arm of the rotatory tube or arm C, from the centre of motion to the centre of the aperture, of any convenient length, not less than one-third (one-ninth according to Mr Gregory,) who has corrected some of Waring's numbers) of the perpendicular height of the water's surface above their centres.

2. Multiply the length of the arm in feet by .614, and take the square root of the product for the proper time of a revolution in seconds, and adapt the other parts of the machinery to this velocity; or, if the time of a revolution be given, multiply the square of this time by 1.63 for the proportional length of the arm.

3. Multiply together the breadth, depth, and velocity per second, of the race, and divide the last product by 18.47 (times 14.27 according to Mr Gregory) the square root of the height, for the area of either aperture.

4. Multiply the area of either aperture by the height of the fall of water, and the product by 41 3/8 pounds (55.775 according to Mr Gregory), for the moving force estimated at the centres of the apertures in pounds avoirdupois.

5. The power and velocity at the aperture may be easily reduced to any part of the machinery by the simplest mechanical rules.

350. Long after the preceding machine had been described in several of our English treatises on machines, Professor Segner published in his hydraulics, as an invention of his own, the account of a machine, differing from this only in form. MN was the axis of the machine, corresponding with DX in Barker's mill, and a number of tubes AB (fig. 94.) were also so arranged round this axis, that their higher extremities A formed a circular superficies, into which the water flowed from a reservoir. When the machine has this form, it has been shewn by Albert Euler that the maximum effect is produced when the velocity is infinite, and that the effect is equal to the power. As a considerable portion of the power, however, must be consumed in communicating to the fluid the circular motion of the tubes; and as the portion thus lost must increase with the velocity of the tube, the effect will in reality sustain a diminution from an increase of velocity.

Sect. II. Description of Albert Euler's Machine driven by the Reaction of the Water.

351. This machine consists of two vessels, the lowest of which EEFF (fig. 95.) is moveable round the vertical axis OO, while the higher vessel remains immovable. The form of the lowest vessel, which is represented by itself in fig. 96, is similar to that of a truncated bell, which is fastened by the cross beams m, n to the axis O, so as to move along with it. The annular cavity hhhh terminates at ee in several tubes ef, ef, ef, diverging from the axis. Through the lower extremities of these tubes, which are bent into a right angle, the water flowing from the cavity hhhh issues with a velocity due to the altitude of its surface in h, h, and produces by its reaction a rotatory and retrograde motion.

---

1 Gregory's Mechanics, vol. ii. p. 111. The cavity of the ring $h$, $h$, receives by the water from the superior vessel GGHH, similar to the inferior vessel in fig. 95, but not connected with the axis OO. This vessel has also an annular cavity PP, into which the water is conveyed from a reservoir by the canal R. Around the lower part HH of the cavity, this vessel is divided into several apertures II, placed obliquely that the water may descend with proper obliquity into the inferior vessel. The width of the higher vessel at HH ought to be equal to the width of the lower vessel at EE, that the water which issues from the former may exactly fill the annular cavity $h$, $h$, $h$.

When the machine is constructed in this way, its maximum effect will be equal to the power, provided all its parts be proportioned and adjusted according to the results in the following table, computed from the formulae of Albert Euler. In the table,

- $Q =$ the quantity of water, or number of cubic feet of water furnished in a second. - $T =$ the time, or number of seconds in which the lower vessel revolves. - $B =$ the breadth of the annular orifice in inches.

### Table for Mills driven by the Reaction of Water.

| Height of the fall of water | Sum of the areas of all the orifices at $f$, $f$, $f$, &c. | Sum of the areas of all the orifices at $f$, $f$, $f$, &c. | Mean radius of the annular orifice HH. | Difference between the altitude of the two vessels. | Tangent of the inclination of the tubes to the horizon. | |-----------------------------|----------------------------------------------------------|----------------------------------------------------------|----------------------------------------|-----------------------------------------------------|-----------------------------------------------------| | Feet. | Square Feet. | Square Inches. | Feet. | Inches. | TTBB | | 1 | 0.17888 $\times Q$ | 2.5759 $\times Q$ | 0.8897 $\times T$ | 1.7695 | 0.38400 $Q$ | | 2 | 0.12649 $\times Q$ | 18.214 $\times Q$ | 1.2592 $\times T$ | 0.8817 | 0.19200 $Q$ | | 3 | 0.103228 $\times Q$ | 14.872 $\times Q$ | 1.5410 $\times T$ | 0.5898 | 0.12800 $Q$ | | 4 | 0.08944 $\times Q$ | 12.880 $\times Q$ | 1.7794 $\times T$ | 0.4424 | 0.09600 $Q$ | | 5 | 0.08000 $\times Q$ | 11.520 $\times Q$ | 1.9894 $\times T$ | 0.3539 | 0.07680 $Q$ | | 6 | 0.07303 $\times Q$ | 10.516 $\times Q$ | 2.1793 $\times T$ | 0.2949 | 0.06400 $Q$ | | 7 | 0.06761 $\times Q$ | 9.736 $\times Q$ | 2.3540 $\times T$ | 0.2528 | 0.05486 $Q$ | | 8 | 0.06325 $\times Q$ | 9.107 $\times Q$ | 2.5165 $\times T$ | 0.2212 | 0.04800 $Q$ | | 9 | 0.05963 $\times Q$ | 8.586 $\times Q$ | 2.6691 $\times T$ | 0.1966 | 0.04267 $Q$ | | 10 | 0.05657 $\times Q$ | 8.146 $\times Q$ | 2.8135 $\times T$ | 0.1769 | 0.03840 $Q$ | | 11 | 0.05394 $\times Q$ | 7.767 $\times Q$ | 2.9508 $\times T$ | 0.1609 | 0.03491 $Q$ | | 12 | 0.05104 $\times Q$ | 7.436 $\times Q$ | 3.0820 $\times T$ | 0.1475 | 0.03200 $Q$ | | 13 | 0.04961 $\times Q$ | 7.144 $\times Q$ | 3.2078 $\times T$ | 0.1351 | 0.02954 $Q$ | | 14 | 0.04781 $\times Q$ | 6.885 $\times Q$ | 3.3290 $\times T$ | 0.1264 | 0.02743 $Q$ | | 15 | 0.04619 $\times Q$ | 6.651 $\times Q$ | 3.4458 $\times T$ | 0.1179 | 0.02560 $Q$ | | 16 | 0.04472 $\times Q$ | 6.440 $\times Q$ | 3.5588 $\times T$ | 0.1106 | 0.02400 $Q$ | | 17 | 0.04339 $\times Q$ | 6.248 $\times Q$ | 3.6683 $\times T$ | 0.1041 | 0.02259 $Q$ | | 18 | 0.04216 $\times Q$ | 6.072 $\times Q$ | 3.7747 $\times T$ | 0.0983 | 0.02133 $Q$ |

The determinations in the preceding table are exhibited in a general manner, that the machine may be accommodated to local circumstances. The time of a revolution $T$, for instance, is left undetermined, because upon this time depends the magnitude of the machine; and $T$ may be assumed of such a value that the dimensions of the machine may be suitable to the given place, or to the nature of the work to be performed.

352. In order to show the application of the preceding table, let it be required to construct the machine when the height of the fall is five feet, and when the reservoir furnishes one cubic foot of water in a second. In this case... On Ma. Q = 1, and therefore, by column 3, the sum of the areas of chimes for the orifices will be 11.52 square inches. Consequently, raising Water.

\[ \frac{11.52}{12} = 0.96 \text{ of a square inch.} \]

Suppose the time of a revolution to be \( t = 1 \) second, or \( T = 1 \), then the 4th column will give the mean radius of the annular orifice \( = 1.9894 \) feet, or nearly two feet. Let the breadth of the annular orifice, or \( B = \frac{1}{2} \) an inch, then the difference between the altitude of each vessel will be \( 0.3539 \times \frac{Q}{TTBB} = \)

\[ 0.3539 \times \frac{1 \times 1}{\frac{1}{2} \times \frac{1}{2} \times 1 \times 1} = 0.3539 \times \frac{1}{\frac{1}{4}} = 0.3539 \times 4 = 1.4156 \text{ inches.} \]

Now, as the sum of the heights of the vessels must be always equal to the height of the fall, half that sum will in the present case be two feet six inches; and since half the difference of their altitudes is 7-10ths of an inch, the altitude of the superior vessel will be two feet six inches and seven-tenths, and that of the inferior vessel two feet five inches and three-tenths. It appears from the last column of the table, that the tangent of the inclination of the tubes is 0.1536, which corresponds with an angle of \( 8^\circ 44' \).

CHAPTER III. ON MACHINES FOR RAISING WATER.

Sect. I. On Pumps.

355. The subject of pumps has been fully and ably discussed by Dr Robison under the article Pump, to which we must refer the reader for a complete view of the theory of the machine. In that article, however, a reference is made to the present for a description of the ancient pump of Ctesibius, and of those in common use to which it has given rise. To these subjects, therefore, we must now confine our attention.

356. The pump was invented by Ctesibius, a mathematician of Alexandria, who flourished under Ptolemy Philopon, about 120 years before Christ. In its original state, it is represented in fig. 97, where ABCD is a brass cylinder with a valve L in its bottom. It is furnished with a piston MK made of green wood, so as not to swell in water, and adjusted to the bore of the cylinder by the interposition of a ring of leather. The tube CI connects the cylinder ABCD with another tube NH, the bottom of which is furnished with a valve I opening upwards. Now, when the extremity DC of the cylinder is immersed in water, and the piston MK elevated, the pressure of the water upon the valve L from below will be proportioned to the depth below the surface (41). The valve will therefore open, and admit the water into the cylinder. But when the piston is depressed, it will force the water into the tube CH, and through the valve I into the tube NH. As soon as the portion of water that was admitted into the cylinder ABCD, is thus impelled into the tube NH, the valve I will close. A second elevation of the piston will admit another quantity of fluid into the cylinder, and a second depression will force it into the tube NH; so that, by continuing the motion of the piston, the water may be elevated to any altitude in the tube. From this pump of Ctesibius are derived the three kinds of pumps now commonly used, the sucking, the forcing, and the lifting pump.

357. The common sucking pump is represented in fig. 98, where ICBL is the body of the pump immersed in the water at A. The moveable piston DG is composed of the piston rod D d, the piston or bucket G, and the valve a. The bucket H, which is fixed to the body of the pump, is likewise furnished with a valve b, which, like the valve a, should by its own weight lie close upon the hole in the bucket till the working of the engine commences. The valves are made of brass, and have their lower surface covered with leather, in order to fit the holes in the bucket more exactly. The moveable bucket G is covered with leather, so as to suit exactly the bore of the cylinder, and to prevent any air escaping between it and the pump. The piston DG may be elevated or depressed by the lever DQ, whose fulcrum is r, the extremity of the bent arm R r.

358. Let us now suppose the piston G to be depressed so that its inferior surface may rest upon the valve b, open. Then, if the piston G be raised to C, there would have been a vacuum between H and G if the valve b were immovable. But as the valve b is moveable, and as the pressure of the air is removed from its superior surface, the air in the tube HI will, by its elasticity, force open the valve b, and expand itself through the whole cavity LC. This air, however, will be much rarer than that of the atmosphere; and since the equilibrium between the external... air and that in the tube LH is destroyed by the rarefaction of the latter, the pressure of the atmosphere on the surface of the water in the vessel K will predominate, and raise the water to about e in the suction pipe HL, so that the air formerly included in the space LC will be condensed to the same state as that of the atmosphere. The elasticity of the air both above and below the valve b being now equal, that valve will fill by its own weight. Let the piston DG be now depressed to b; the air would evidently resist its descent, did not the valve a open and give a free exit to the air in the space CH, for it cannot escape through the inferior valve b. When the piston reaches b, the valve a will fall by its weight; and when the piston is again elevated, the incumbent air will press the valve a firmly upon its orifice. During the second ascent of the piston to C, the valve b will rise, the air between eH will rush into HC; and in consequence of its rarefaction, and inability to counteract the pressure of the atmosphere, the water will rise to f. In the same way it may be shewn, that at the next stroke of the piston the water will rise through the box H to B; and then the valve b which was raised by it will fall when the bucket G is at C. Upon depressing the bucket G again, the water cannot be driven through the valve b, which is pressed to its orifice by the water above it. At the next ascent of the piston a new quantity of water will rise through H, and follow the piston to C. When the piston again descends, the valve a will open; and as the water between C and H cannot be pushed through the valve b, it will rise through a, and have its surface at C when the piston G is at b; but when the piston rises, the valve a being shut by the water above it, this water will be raised up towards I, and issue at the pipe F. A new quantity of water will rush through H, and fill the space HC; consequently, the surface of the fluid will always remain at C, and every succeeding elevation of the piston from b to C will make the column of water CH run out at the pipe F.

359. As the water rises in the pipe CL solely by the pressure of the atmosphere; and as a column of water, 33 feet high, is equal in weight to a column of air of the same base, reaching from the earth's surface to the top of the atmosphere, the water in the vessel K will not follow the piston G to a greater altitude than 33 feet; for when it reaches this height, the column of water completely balances, or is in equilibrium with, the atmosphere, and therefore cannot be raised higher by the pressure of the external air.

360. The forcing-pump is represented in fig. 99, where D d is the piston attached to a solid plunger g, adjusted to the bore of the pipe BC by the interposition of a ring of leather. The rectangular pipe MMN communicates with the tube BC by the cavity round H; and its upper extremity P is furnished with a valve a opening upwards. An air-vessel KK is fastened to P, and the tube EGI is introduced into it so as to reach as near as possible to the valve a. Let us now suppose the plunger D g to be depressed to b. As soon as it is elevated to C, the air below it will be rarefied, and the water will ascend through the valve b in the same way as in the sucking-pump, till the pipe is filled to C. The valve b will now be shut by the weight of the incumbent water; and therefore, when the plunger D g is depressed, it will force the water between C and b through the rectangular pipe MMN, into the air vessel KK. Before the water enters the air-vessel, it opens the valve a, which shuts as soon as the plunger is again raised, because the pressure of the water upon its under side is removed. In this way the water is driven into the air-vessel by repeated strokes of the plunger, till its surface is above the lower extremity of the pipe IG. Now, as the air in the vessel KK has no communication with the external air when the water is above I, it must be condensed more and more, as new quantities of water are injected. It will therefore endeavour to expand itself; and by pressing upon the surface H of the water in the air-vessel, it will drive the water through the tube IG, and make it issue at E in a continued stream, even when the plunger is rising to C. If the pipe GHI were joined to the pipe MMN at P, without the intervention of an air vessel, the stream of water would issue at F only when the plunger was depressed.

361. The lifting pump, which is only a particular modification of the forcing pump, is represented in fig. 100. The barrel AB is fixed in the immovable frame KILM, the lower part of which is immersed in the water to be raised. The frame GEQHO consists of two strong iron rods EQ, OH, which move through holes in IK and LM, the upper and lower ends of the pump. To the bottom QH of this frame is fixed an inverted piston, with its bucket and valve uppermost at D. An inclined branch FH, either fixed to the top of the barrel, or moveable by a ball and socket, as represented at F, must be fitted to the barrel so exactly, as to resist the admission both of air and water. The branch FH is furnished with a valve C opening upwards. Let the pump be now plunged in the water to the depth of D. Then if the piston frame be thrust down into the fluid, the piston will descend, and the water by its upward pressure will open the valve at D, and gain admission above the piston. When the piston frame is elevated, it will raise the water above D along with it, and, forcing it through the valve, it will be carried off by the spout above H.

362. An ingenious pump, invented by De la Hire, is represented in fig. 101. It raises water equally quick by Hire's the descent as by the ascent of the piston. The pipes B, C, E, F, all communicate with the barrel MD, and have each a valve at their top, viz. at b, S, e, f. The piston rod LM and plunger K never rise higher than K, nor descend lower than D, KD being the length of the stroke. When the plunger K is raised from D to K, the pressure of the atmosphere forces the water through the valve b, and fills the barrel up to the plunger, in the very same way as in the forcing pump. When the plunger K is depressed to D, it forces the water between K and b up the pipe E, and through the valve e into the box G, where it issues at the spout O. During the descent of the plunger K, the valve f falls, and covers the top of the pipe F; and as the piston rod LM moves in a collar of leather at M, and is air-tight, the air above the plunger, between K and M, will be rare- On Machines for raising Water.

Fig. 101.

Noble's pump.

Fig. 102.

Fig. 103.

fied, and likewise the air in the pipe CS, which communicates with the rarefied air by the valve S. The pressure of the air, therefore, will raise the water in CS, force it through the valve S, and fill the space above the plunger, expelling the rarefied air through the valve f. When the piston is raised from D to K, it will force the water through the bent pipe F into the box G, so that the same quantity of water will be discharged at O through the pipe F, during the ascent of the piston, as was discharged through the pipe E during the piston's descent. Above the pipe O is a close air-vessel P, so that when the water is driven above the spout O, it compresses the air in the vessel P, and this air acting by its elasticity on the surface of the water, forces it out at O in a constant and nearly equal stream. As the effect of the machine depends on a proper proportion between the height O of the spout above the surface of the well, and the diameter of the barrel, the following table will be of use to the practical mechanic.

| Height of the spout O above the well. | Diameter of the barrel D. | |--------------------------------------|--------------------------| | Feet. | Inches. | | 10 | 6.9 | | 15 | 5.6 | | 20 | 4.9 | | 25 | 4.4 | | 30 | 4.0 | | 35 | 3.7 | | 40 | 3.5 | | 45 | 3.3 | | 50 | 3.1 |

Height of the spout O above the well.

| Diameter of the barrel D. | |---------------------------| | Inches. | | 60 | | 65 | | 70 | | 75 | | 80 | | 85 | | 90 | | 95 | | 100 |

When the proportions in the preceding table are observed, a man of common strength will raise water much higher than he could do with a pump of the common construction.

363. A very simple pump, which furnishes a continued stream, is represented in fig. 102. It was invented by Mr Buchanan.

364. The pump invented by Mr Buchanan is shewn in fig. 103. In the vertical section DGA, A is the suction barrel, D the working barrel, E the piston, G the spout, B the inner valve, and C the outer valve. These valves are of the kind called clack valves, and have their hinges generally of metal. It is easily seen that when the piston E is raised, the water will rise through the suction barrel A into the working barrel D, in the same way as in the sucking pump; and that when the piston E is depressed, it will force the water between it and the valve B, through the valve C, and make it issue at G. The points of difference between this pump and those of the common form, are,—that it discharges the water below the piston, and has its valves lying near each other. Hence the sand or mud which may be in the water, is discharged without injuring the barrel or the piston leathers; and as the valves B, C may be of any size, they will transmit, without being choked, any rubbish which may rise in the suction barrels. If any obstruction should happen to the valves, they are within the reach of the workman's hand, and may be cleared without taking the pump to pieces. This simple machine may be quickly converted into a fire-engine, by adding the air-vessel H, which is screwed like a hosepipe, and by fixing in the spout G, a perforated stopple fitted to receive such pipes as are employed in fire-engines. When these additions are made, the water, as in the case of the forcing pump, will be driven into the air vessel H, and repelled through the perforated stopple G, by the elasticity of the included air.

365. A simple method of working two pumps at once, by means of a balance, is exhibited in fig. 104, where AB is the balance, having a large iron ball at each end, placed in equilibrium on the two spindles C, see fig. 105. The person who works the pump stands on two boards I, I, nailed to two cross pieces fastened to the axis of the machine, and supports himself by a cross bar D joined to the two parts D, E. At the distance of ten inches on each side of the axis are suspended the iron rods M, N, to which the pistons are attached. The workman, by bearing alternately on the right and left foot, puts the balance in motion. The pistons M, N, are alternately elevated and depressed, and the water raised in the barrel of each is driven... into the pipe HH, in which it is elevated to a height proportional to the diameter of the valves, and the power of the balance. In order to make the oscillations of the balance equal, and prevent it from acquiring too great a velocity, iron springs F, G, are fixed to the upright posts, which limit the length of its oscillations.

366. The chain pump is represented in fig. 106. It consists of a chain MTHG, about 30 feet long, carrying a number of flat pistons M, N, O, P, Q, which are made to revolve in the barrels ABCD and GH, by driving the wheel F. When the flat pistons are at the lower part of the barrel T, they are immersed in the water RR, and as they rise in the barrel GH, they bring up the water along with them into the reservoir MG, from which it is conveyed by the spout S. The teeth of the wheel F are so contrived as to receive one-half of the flat pistons, and let them fold in; and sometimes another wheel like F is fixed at the bottom D. The distance of the pistons from the side of the barrel is about half an inch; but as the machine is generally worked with great velocity, the ascending pistons bring along with them into the reservoir as much water as fills the cavity GH. Sometimes chain pumps are constructed without the barrels ABCD and GH. In this case, the flat pistons are converted into buckets connected with a chain, which dip in the water with their mouths downwards, and convey it to the reservoir. The buckets are moved by hexagonal axles, and the distance between each is nearly equal to the depth of the buckets. Chain pumps are frequently placed in an inclined position, and in this position they raise the greatest quantity of water when the distance of the flat pistons is equal to their breadth, and when the inclination of the barrels is about 24° 21'.

367. The hair-rope machine, invented by the Sieur Ve-Hair-rope ra, operates on the same principle as the chain-pump. Instead of a chain of pistons moving round the wheel F (fig. 106), a hair-rope is substituted. The part of the rope at T that is lowest always dips in the water, which, adhering to the rope, is raised along with it. When the rope reaches the top at G and M, it passes through two small tubes, which, being fixed in the bottom of the reservoir, prevent the water from returning into the well. Sometimes a common rope is employed, having a number of stuffed cushions fixed to it instead of the flat pistons in the chain-pump. These cushions carry the water along with them through the barrel HG, and deliver it into the reservoir.—For the description of other pumps, see the article Pump; and for pump mills, see the article Mill.

Sect. II. On Engines for Extinguishing Fire.

368. The common fire-engine which discharges water in successive jets is represented in fig. 107, and is only a squirting engine.

Fig. 107.

In the vessel AB, full of water, is immersed the frame DC of a common lifting pump. This frame, and consequently the piston N, is elevated and depressed by means of the levers E, F, and the water which is raised is forced through the pipe G, which may be moved in any direction by means of the elastic leather pipe H, or by a ball and socket screwed on the top of the pump. While the piston N is descending, the stream at G is evidently discontinued, and issues only at each elevation of the piston. The vessel AB is supplied with water by buckets, and the pump is prevented from being choked by the strainer LK, which separates from the water any mud that it may happen to contain.

369. As this fire-engine does not afford a continued stream, it is not so useful in case of accidents as when the fire-engine stream is uninterrupted. An improved engine of this sort is represented in fig. 108, where H, H, are two forcing pumps connected with the large vessel LMM, and wrought by the levers X, X, moving upon Y as a fulcrum. apparatus is plunged and fastened in a vessel II partly filled with water, and by means of the forcing pumps H, H, the operation of which has already been described, the water is raised through the valves above II, and driven through the valves A, A into the large vessel LMM, where the included air is condensed. Into this vessel is inserted the tube YY, communicating with the leathern hose which carries off the water. The elasticity of the condensed air in the vessel LMM pressing upon the surface of the water in that vessel, forces it up through the tube YY into the leathern pipe, from whose extremity it issues with great force and velocity; and as the condensed air is continually pressing upon the water in the air-vessel, the stream will be constant and uniform.

Newsham's fire-engine, as improved by Mr Newsham, is represented in fig. 109, where TU and WX are the forcing pumps corresponding with H, H in fig. 108, YZ the large vessel corresponding with LMM, and ef the tube corresponding with YY. The vessels TU, WX, YZ, the horizontal canals ON, QP, ML, and the vertical canal EE, all communicate with each other by means of four valves O, I, K, P opening upwards, and the vertical pipe is immersed in the water to be raised. When the piston R is raised by means of the double lever αβ, a vacuum would be made in the barrel TU, if the water at R were prevented from rising; but as this barrel communicates with the vessel of water below EF, on the surface of which the pressure of the atmosphere is exerted, the water will rise through EF, force open the valve H, and follow the piston R. By depressing the piston R, however, the water is driven down the barrel, closes the valve H, and rushes through the valve I into the air-vessel YZ. The very same operation is going on with the pump WX, which forces the water into the air-vessel through the valve K. By these means the air-vessel is constantly filling with water, and the included air undergoing continual condensation. The air thus compressed, reacts upon the surface YZ of the water, and forces it through the tube ef to the stop-cock eg, whence, after turning the cock, the water passes into the tube h, fixed to a ball and socket, by which it may be discharged in any direction.

371. The fire-engine has undergone various alterations and improvements from Bramah, Dickenson, Simpkin, Ra-ventreé, Philips, and Furst, an account of whose engines may be seen in the Repertory of Arts, &c. A very simple and cheap fire-engine has been invented by Mr B. Dearborn, and is described in the American Transactions for 1794, and in Gregory's Mechanics, vol. ii. p. 177.

Sect. III. On Whitehurst's Machine, and Montgolfier's Hydraulic Ram.

372. Mr Whitehurst was the first who suggested the ingenious idea of raising water by means of its momentum, but in an improved form, has lately made its appearance in France, and excited considerable attention both on the Continent and in this country. Whatever credit, therefore, has been given to the inventor of the hydraulic ram, justly Mr W. belongs to our countrymen Mr Whitehurst, and Montgolfier is entitled to nothing more than the merit of an improver.

373. Mr Whitehurst's machine, which was actually erected at Oulton in Cheshire, is represented in fig. 110, where AM is the original reservoir, having its surface in the same horizontal line with the bottom of the reservoir BN. The diameter of the main pipe AE is one inch and a half, and its length about 200 yards; and the branch pipe EF is of such a size that the height of the surface M of the reservoir is nearly 16 feet above the cock F. In the valve box D is placed the valve a, and into the air-vessel C are inserted the extremities m, n of the main pipe, bent downwards to prevent the air from being driven out when the water is forced into it. Now as the cock F is 16 feet below the reservoir AM, the water will issue from F with a velocity

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1 Philosophical Transactions, 1775. of nearly 30 feet per second. As soon as the cock F therefore is opened, a column of water 200 yards long is put in motion, and though the aperture of the cock F be small, this column must have a very considerable momentum. Let the cock F be now suddenly stopped, and the water will rush through the valve a into the air-vessel C, and condense the included air. This condensation must take place every time the cock is shut, and the imprisoned air being in a state of high compression, will react upon the water in the air-vessel, and raise it into the reservoir BN.

374. A section of the hydraulic ram of Montgolfier is exhibited in fig. 111, where X is the reservoir, XA the height of the fall, and AB the horizontal canal which conveys the water to the engine MFKR. C and D are two valves, and ME a pipe reaching within a very little of the bottom E. Let us now suppose that water is permitted to descend from the reservoir. It will evidently rush along AB, and out at HH, which can be shut by the valves C or D. When the passage HH is shut by the rise of the valve C, the water is suddenly checked, and, unable to escape at HH, it will rush forwards to Z and raise the valves at E. A portion of water being thus admitted into the vessel FF, the impulse of the column of fluid is spent, the valves D and C fall, and the water issues at HH as before; when its motion is again checked, and the same operation repeated which has now been described. Whenever, therefore, the valve C closes, a portion of water will force its way into the vessel FF, and condense the air which it contains, for the included air has no communication with the atmosphere after the bottom of the pipe ME is beneath the surface of the injected water. This condensed air will consequently react upon the surface of the water, and raise it in the pipe ME to an altitude proportioned to the elasticity of the included air.

An enlarged view of the valves C, D, and their seat, is shewn in the annexed figure (fig. 112). With a fall of water in which XA is five feet, the pipe AB six inches in diameter, and 14 feet long, a good proportioned ram will raise water 100 feet high. When wrought with a power of 70 cubic feet of water in the minute, it will raise 2½ cubic feet of water per minute to the height of 100 feet. One of these machines is stated to have raised 100 hogsheads of water in 24 hours to the height of 134 feet, and a fall of only 4½ feet. From this description it will be seen, that the only difference between the engines of Montgolfier and Whitehurst is, that the one requires a person to turn the cock, while the other has the advantage of acting spontaneously. Montgolfier assumes us, that the honour of this invention does not belong to England, but that he is the sole inventor, and did not receive a hint from any person whatever.—It would appear from some experiments made by Montgolfier, that the effect of the water-ram is equal to between a half and three-fourths of the power expended, which renders it superior to most hydraulic machines.

Sect. IV. On Archimedes's Screw-Engine.

375. The screw-engine invented by Archimedes is represented in fig. 113, where AB is a cylinder with a flexible pipe CEHOGF, wrapped round its circumference like a chimes's screw. The cylinder is inclined to the horizon, and supported at one extremity by the bent pillar IR, while its other extremity, furnished with a pivot, is immersed in the water. When, by means of the handle K, the cylinder is made to revolve upon its axis, the water which enters the lower orifice of the flexible pipe is raised to the top, and discharged at D. On some occasions, when the water to be raised moves with a considerable velocity, the engine is put in motion by a number of floatboards fixed at L, and impelled by the current; and if the water is to be raised to a great height, another cylinder is immersed in the vessel D, which receives the water from the first cylinder, and is driven by a pinion fixed at I. In this way, by having a succession of screw-engines, and a succession of reservoirs, water may be raised to any altitude. An engine of this kind is described in Ferguson's Lectures, vol. ii. p. 113.

376. In order to explain the reason why the water rises in the spiral tube, let AB be a section of the engine, BCDE the spiral tube, BF a horizontal line or the surface of the stagnant water which is to be raised, and ABF the angle which the axis of the cylinder makes with the horizon. Then, the water which enters the extremity B of the spiral tube will descend to C, and remain there as long as the cylinder is at rest. But if a motion of rotation be communicated to the cylinder, so that the lowest part C of the spiral BCD moves towards B, and the points d, D, E towards C, and become successively the lowest parts of the spiral, the water must occupy successively the

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1 Cette invention n'est point originale d'Angleterre, elle appartient toute entière à la France. Je déclare que j'en suis le seul inventeur, et que l'idée ne m'en a été fournie par personne. Journal des Mines, vol. xiii. No. 73. 2 See Appendix to Ferguson's Lectures; Repertory of Arts, Dec. 1816; and Journal of the Royal Institution, vol. i. p. 211. On Machines for raising water.

376. Points \(d\), \(D\), \(E\), and therefore rise in the tube; or, which is the same thing, when the point \(C\) moves to \(e\), the point \(d\) will be at \(C\); and as the water at \(C\) cannot rise along with the point \(C\) to \(e\), on account of the inclination of \(Ce\) to the horizon, it must occupy the point \(d\) of the spiral, when \(C\) has moved to \(e\); that is, the water has a tendency to occupy the lower parts of the spiral, and the rotatory motion withdraws this part of the spiral from the water, and causes it to ascend to the top of the tube. By wrapping a cord round a cylinder, and inclining it to the horizon, so that the angle \(ABC\) may be greater than the angle \(ABF\), and then making it revolve upon its axis, the preceding remarks will be clearly illustrated.—If the direction of the spiral \(BC\) should be horizontal, that is, if it should coincide with the line \(BF\), the water will have no tendency to move towards \(C\), and therefore cannot be raised in the tube. For a similar reason, it will not rise when the point \(C\) is above the horizontal line \(BF\). Consequently, in the construction of this engine, the angle \(ABC\), which the spiral forms with the side of the cylinder, must always be greater than the angle \(ABF\), at which the cylinder is inclined to the horizon. In practice, the angle of inclination \(ABF\) should generally be about \(50^\circ\), and the angle \(ABC\) about \(65^\circ\).

377. The screw of Archimedes is now generally constructed as shewn in the annexed figure, where \(AB\) is the axis of the screw, having a flat plate of wood or thin iron coiled, as it were, round the axis, like a spiral, or the threads of a screw. The plane of this plate is perpendicular to the surface of the cylindrical axis \(AB\), but is inclined to the direction of the axis at an angle which must always be greater than the angle which the axis \(AB\) forms with the horizon when in use. This spiral plate, which is nothing more than a wooden screw with a very deep and narrow thread, is fixed in a cylindrical box, \(CDEF\), so as to form a spiral groove, as it were running up the tube from \(B\) to \(A\), which is exactly the same thing as if a pipe of lead or leather had been wrapped round the cylindrical axis, as in fig. 115. If the outer case \(CDEF\) is fixed so that the screw revolves within it, the engine is called a water screw-engine. In the common screw-engine, Mr Eytelwein has shewn that the screw should be placed in such a manner that only one-half of a convolution may be filled at each revolution. When this condition, however, cannot be fulfilled, from the height of the water being variable, he gives a preference to the water screw, notwithstanding that in this case one-third of the water generally runs back, and the screw is apt to become clogged by impurities or weeds.

378. In a screw-engine erected at the Hurlet Alum Works, for raising the alum liquor, the length of the screw is 127 feet, its inclination to the horizon \(37^\circ 36'\); the height to which it raises the liquor 76 feet 9 inches, the octagonal axis of the screw 8 inches in diameter, the diameter of the spiral 22 inches, the thickness of the covering 2 inches, the distance of the threads 9 inches, the number of the threads 168, the thickness of the spiral 2 inches, the width and depth of the spiral tube 7 inches each. The screw is sustained upon five sets of pivots or rollers, each set consisting of two rollers. The engine is driven by a water-wheel, which performs one revolution while the screw performs two. The quantity of liquor raised is 70 wine gallons; and as its specific gravity is 1.065, the quantity discharged in an hour is 17 tons. The screw is built wholly of wood, as the alum liquor acts upon iron.

379. The theory of this engine is treated at great length by Hennert, in his Dissertation sur la vis d'Archimède, Berlin, 1767; by Pitot, in the Memoirs of the French Academy; and by Euler, in the Nov. Comment. Petrop. tom. v. An account of Pitot's investigations may be seen in Gregory's Mechanics, vol. ii. p. 348. See also Eytelwein's Handbuch der Mechanik, ch. xxii.; and Journal des Mines, tom. xxxviii. p. 321.

Sect. V. On the Persian Wheel.

380. The Persian wheel is an engine which raises water to a height equal to its diameter. It is shewn in fig. 116, where \(AB\) is the wheel driven by the stream \(r\) acting upon the boatboards fixed on one side of its rim. A number of buckets, \(n, o, p, q\), are disposed on the opposite side of the rim, and suspended by strong pins. When the wheel is in motion, the descending buckets \(q, q\) immerge into the stream, and ascend full of water till they reach the top at \(p\), where they strike against the extremity of the fixed reservoir \(S\), and being overset, discharge their contents into that reservoir. As soon as the bucket quits the reservoir, it resumes its perpendicular position by its own weight and descends as before. On each bucket is fixed a spring \(t, t\), which moves over the top of the bar fastened to the reservoir \(S\). By this means the bottom of the bucket is raised above the level of its mouth, and its contents completely discharged.

381. On some occasions the Persian wheel is made to raise water only to the height of its axle. In this case, instead of buckets, its spokes, C, C, C, &c., are made of a spiral form, and hollow within, so that their inner extremities all terminate in the box N on the axle, and their outer extremities in the circumference of the wheel. When the rim AB, therefore, is immersed in the stream, the water runs into the tubes I, I, &c., rises in the spiral spokes C, C, C, &c., and is discharged from the orifices within m into the reservoir N, from which it may be conveyed in pipes.

Sect. VI. On the Zurich Machine.

382. This machine is a kind of pump invented and erected by H. Andreas Wirtz, an ingenious tin-plate worker in Zurich, and operates on a principle different from all other hydraulic engines. The following description of it, written by Dr Robison, is transferred to this part of the work for the sake of uniformity.

383. Fig. 117. is a sketch of the section of the machine, as it was first erected by Wirtz at a dye-house in Limmat, in the suburbs or vicinity of Zurich. It consists of a hollow cylinder, like a very large grindstone, turning on a horizontal axis, and partly plunged in a cistern of water. The axis is hollow at one end, and communicates with a perpendicular pipe CBZ, part of which is hid by the cylinder. This cylinder or drum is formed into a spiral canal by a plate coiled up within it like the main-spring of a watch in its box; only the spires are at a distance from each other, so as to form a conduit for the water of uniform width. This spiral partition is well joined to the two ends of the cylinder, and no water escapes between them. The outermost turn of the spiral begins to widen about three-fourths of a circumference from the end, and this gradual enlargement continues from Q to S nearly a semicircle: this part may be called the Horn. It then widens suddenly, forming a Scoop or shovel SS'. The cylinder is supported so as to dip several inches into the water, whose surface is represented by VV'.

384. When this cylinder is turned round its axis in the direction ABEQ, as expressed by the two darts, the scoop SS' dips at V' and takes up a certain quantity of water before it immerses again at V. This quantity is sufficient to fill the taper part SQ, which we have called the Horn; and this is nearly equal in capacity to the outermost uniform spiral round.

385. After the scoop has emerged, the water passes along the spiral by the motion of it round the axis, and drives the air before it into the rising-pipe, where it escapes.—In the mean time, air comes in at the mouth of the scoop; and chimes for raising Water.

On Ma. On Machines for raising Water.

The gradual diminution of the water columns will be produced during the motion by the water running over backwards at the top, from spire to spire, and at last coming out by the scoop.

388. It is evident that this disposition of the air and water will raise the water to the greatest height, because the hydrostatic height of each water column is the greatest possible, viz. the diameter of the spire. This disposition may be obtained in the following manner: Take CL to CB as the density of the external air to its density in the last column next the rising pipe or main; that is, make CL to CB as 33 feet (the height of the column of water which balances the atmosphere), to the sum of 33 feet and the height of the rising pipe. Then divide BL into such a number of turns, that the sum of their diameters shall be equal to the height of the main; then bring a pipe straight from L to the centre C. The reason of all this is very evident.

389. But when the main is very high, this construction will require a very great diameter of the drum, or many turns of a very narrow pipe. In such cases it will be much better to make the spiral in the form of a cork-screw, as in fig. 118, instead of this flat form like a watch-spring. The pipe which forms the spiral may be lapped round the frustum of a cone, whose greatest diameter is to the least (which is next to the rising pipe) in the same proportion that we assigned to CB and CL. By this construction the water will stand in every round so as to have its upper and lower surfaces tangents to the top and bottom of the spiral, and the water columns will occupy the whole ascending side of the machine, while the air occupies the descending side.

390. This form is vastly preferable to the flat one; it will allow us to employ many turns of a large pipe, and therefore produce a great elevation of a large quantity of water.

The same thing will be still better done by lapping the pipe on a cylinder, and making it taper to the end, in such a proportion that the contents of each round may be the same as when it is lapped round the cone. It will raise the water to a greater height (but with an increase of the impelling power) by the same number of turns, because the vertical or pressing height of each column is greater.

Nay, the same thing may be done in a more simple manner, by lapping a pipe of uniform bore round a cylinder. But this will require more turns, because the water columns will have less differences between the heights of their two ends. It requires a very minute investigation to show the progress of the columns of air and water in this construction, and the various changes of their arrangement, before one is attained which will continue during the working of the machine.

391. We have chosen for the description of the machine that construction which made its principle and manner of working most evident; namely, which contained the same material quantity of air in each turn of the spiral, more and more compressed as it approaches to the rising pipe. We should otherwise have been obliged to investigate in great detail the gradual progress of the water, and the frequent changes of its arrangement, before we could see that one arrangement would be produced which would remain constant during the working of the machine. But this is not the best construction. We see that, in order to raise water to the height of a column of 34 feet, which balances the atmosphere, the air in the last spire is compressed into half its bulk; and the quantity of water delivered into the main at each turn is but half of what was received into the first spire, the rest flowing back from spire to spire, and being discharged at the spout.

392. But it may be constructed so as that the quantity of water in each spire may be the same that was received into the first; by which means a greater quantity (double in the instance now given) will be delivered into the main, and raised to the same height by very nearly the same force.—This may be done by another proportion of the capacity of the spires, whether by a change of their caliber or of their diameters. Suppose the bore to be the same, the diameter must be made such that the constant column of water, and the column of air, compressed to the proper degree, may occupy the whole circumference. Let A be the column of water which balances the atmosphere, and h the height to which the water is to be raised. Let A be to \(A + h\) as 1 to \(m\).

393. It is plain that \(m\) will represent the density of the air in the last spire, if its natural density be 1, because it is pressed by the column \(A + h\), while the common air is pressed by A. Let 1 represent the constant water column, and therefore nearly equal to the air column in the first spire. The whole circumference of the last spire must be

\[ 1 + \frac{1}{m} \]

in order to hold the water 1, and the air compressed into the space

\[ \frac{1}{m} \text{ or } \frac{A}{A + h}. \]

394. The circumference of the first spire is \(1 + 1\) or 2. Let D and d be the diameters of the first and last spires; we have

\[ 2 : 1 + \frac{1}{m} = D : d, \text{ or } 2m : m + 1 = D : d. \]

Therefore, if a pipe of uniform bore be lapped round a cone, of which D and d are the end diameters, the spirals will be very nearly such as will answer the purpose. It will not be quite exact, for the intermediate spirals will be somewhat too large. The conoidal frustum should be formed by the revolution of a curve of the logarithmic kind. But the error is very trifling.

With such a spiral, the full quantity of water which was confined in the first spiral will find room in the last, and will be sent into the main at every turn. This is a very great advantage, especially when the water is to be much raised. The saving of power by this change of construction is always in proportion to the greatest compression of the air.

The great difficulty in the construction of any of these forms is in determining the form and position of the horn and the scoop; and on this greatly depends the performance of the machine. The following instructions will make it pretty easy.

395. Let ABEO (fig. 119.) represent the first or outermost round of the spiral, of which the axis is C. Suppose it immersed up to the axis in the water VV, we have seen that the machine is most effective when the surfaces KB and ON of the water columns are distant the whole diameter BO of the spiral. Therefore, let the pipe be first supposed of equal calibre to the very mouth E e, which we suppose to be just about to dip into the water. The sur- the quadrant OE, and in the quadrant which lies behind EB. And this compression is supported by the columns behind, between this spire and the rising pipe. But the air in the outermost quadrant EB is in its natural state, communicating as yet with the external air. When, however, the mouth E has come round to A, it will not have the water standing in it in the same manner, leaving the half space BEO filled with compressed air; for it took in and confined only what filled the quadrant BE. It is plain, therefore, that the quadrant BE must be so shaped as to take in and confine a much greater quantity of air; so that when it has come to A, the space BEO may contain air sufficiently dense to support the column AO. But this is not enough: for when the wide mouth, now at A α, rises up to the top, the surface of the water in it rises also, because the part Ao O α is more capacious than the cylindric part OE e o which succeeds it, and which cannot contain all the water that it does. Since, then, the water in the spire rises above A, it will press the water back from O n to some other position m' n', and the pressing height of the water-column will be diminished by this rising on the other side of O. In short, the horn must begin to widen, not from B but from A, and must occupy the whole semicircle ABE; and its capacity must be to the capacity of the opposite cylindrical side as the sum of BO, and the height of a column of water which balances the atmosphere to the height of that column. For then the air which filled it, when of the common density, will fill the uniform side BEO, when compressed so as to balance the vertical column BO.

But even this is not enough: for it has not taken in enough of water. When it dipped into the cistern at E, it carried air down with it, and the pressure of the water in the cistern caused the water to rise into it a little way; and some water must have come over at B from the other side, which was drawing narrower. Therefore, when the horn is in the position BOA, it is not full of water. Therefore, when it comes into the situation OAB, it cannot be full nor balance the air on the opposite side. Some will therefore come out at O, and rise up through the water. The horn must, therefore, 1st, Extend at least from O to B, or occupy half the circumference; and, 2dly, It must contain at least twice as much water as would fill the side BEO. It will do little harm though it be much larger; because the surplus of air which it takes in at E will be discharged, as the end E e of the horn rises from O to B, and it will leave the precise quantity that is wanted. The overplus water will be discharged as the horn comes round to dip again into the cistern. It is possible, but requires a discussion too intricate for this place, to make it of such a size and shape, that while the mouth moves from E to B, passing through O and A, the surface of the water in it shall advance from E to O n, and be exactly at O when the beginning or narrow end of the horn arrives there.

400. We must also secure the proper quantity of water. When the machine is so much immersed as to be up to the axis in water, the capacity which thus secures the proper quantity of air will also take in the proper quantity of water. But it may be erected so as that the spirals shall not even reach the water. In this case it will answer our purpose if we join to the end of the horn a scoop or shovel QRSB (fig. 120.), which is so formed as to take in at least as much water as will fill the horn. This is all that is wanted in the beginning of the motion along the spiral, and more than is necessary when the water has advanced to the succeeding spire; but the overplus is discharged in the way we have mentioned. At the same time, it is needless to load the machine with more water than is necessary, merely to throw it out again. We think that if the horn occupies fully more than one-half of the circumference, and contains as much as will fill the whole round, and if the scoop lifts as much as will certainly fill the horn, it will do very well.

The scoop must be very open on the side next the axis, that it may not confine the air as soon as it enters the water. This would hinder it from receiving water enough.

401. The following dimensions of a machine erected at Florence, and whose performance corresponded extremely well with the theory, may serve as an example.

The spiral is formed on a cylinder of ten feet diameter, and the diameter of the pipe is six inches. The smaller end of the horn is of the same diameter; it occupies three-fourths of the circumference, and is 7½ inches wide at the outer end. Here it joins the scoop, which lifts as much water as fills the horn, which contains 4340 Swedish cubic inches, each = 1.577 English. The machine makes six turns in a minute, and raises 1354 pounds of water, or 22 cubic feet, 10 feet high in a minute.

402. The above account will, we hope, sufficiently explain the manner in which this singular hydraulic machine produces its effect. When every thing is executed by the maxims which we have deduced from its principles, we are confident that its performance will correspond to the theory; and we have the Florentine machine as a proof of this. It raises more than ten-elevenths of what the theory promises, and it is not perfect. The spiral is of equal caliber, and is formed on a cylinder. The friction is so inconsiderable in this machine, that it need not be minded; but the great excellency is, that whatever imperfection there may be in the arrangement of the air and water columns, this only affects the elegance of the execution, causing the water to make a few more turns in the spiral before it can mount to the height required; but wastes no power, because the power employed is always in proportion to the sum of the vertical columns of water in the rising side of the machine; and the height to which the water is raised by it is in the very same proportion. It should be made to move very slow, that the water be not always dragged up by the pipes, which would cause more to run over from each column, and diminish the pressure of the remainder.

403. If the rising-pipe be made wide, and thus room be made for the air to escape freely up through the water, it will rise to the height assigned; but if it be narrow, so that the air cannot get up, it rises almost as slow as the water, and by this circumstance the water is raised to a much greater height mixed with air, and this with hardly any more power. It is in this way that we can account for the great performance of the Florentine machine, which is almost triple of what a man can do with the finest pump that ever was made; indeed, the performance is so great, that one is apt to suspect some inaccuracy in the accounts.

The entry into the rising-pipe should be no wider than the last part of the spiral; and it would be advisable to divide it into four channels by a thin partition, and then to make the rising-pipe very wide, and to put into it a number of slender rods, which would divide it into slender channels that would completely entangle the air among the water. This will greatly increase the height of the heterogeneous column. It is surprising that a machine that is so very promising should have attracted so little notice. We do not know of any being erected out of Switzerland, except at Florence in 1778. The account of its performance was in consequence of a very public trial in 1779, and honourable declaration of its merit, by Sig. Lorenzo Ginori, who erected another, which fully equalled it. It is shortly mentioned by Professor Sulzer of Berlin, in the Sammlungen Vermischten Schriften for 1754. A description of it is published by the Philosophical Society at Zurich in 1766, and in the descriptions published by the Society in London for the encouragement of Arts in 1776. The celebrated Daniel Bernoulli has published a very accurate theory of it in the Petersburgh Commentaries for 1772, and the machines at Florence were erected according to his instructions. Baron Alstromer in Sweden caused a glass model of it to be made, to exhibit the internal motions for the instruction of artists, and also ordered an operative engine to be erected; but we have not seen any account of its performance. It is a very intricate machine in its principles; and an ignorant engineer, nay the most intelligent, may erect one which shall hardly do any thing; and yet by a very trifling change, may become very powerful. We presume that failures of this kind have turned the attention of engineers from it; but we are persuaded that it may be made very effective, and we are certain that it must be very durable. Fig. 121. is a section of the manner in which the author has formed the communication between the spiral and the rising-pipe. P is the end of the hollow axis which is united with the solid iron axis. Adjoining to P, on the under side, is the entry from the last turn of the spiral. At Q is the collar which rests on the supports, and turns round in a hole of bell-metal. ff is a broad flanch cast in one piece with the hollow part. Beyond this the pipe is turned somewhat smaller, very round and smooth, so as to fit into the mouth of the rising-pipe, like the key of a cock. This mouth has a plate e attached to it. There is another plate dd, which is broader than ee, and is not fixed to the cylindrical part, but moves easily round it. In this plate are four screws, such as gg, which go into holes in the plate ff, and thus draw the two plates ff and dd together, with the plate ee between them. Pieces of thin leather are put on each side of ee; and thus all escape of water is effectually prevented, with a very moderate compression and friction.

CHAPTER IV. ON MACHINES IN WHICH WATER IS THE CHIEF AGENT.

Sect. I. On the Water-Blowing Machine.

404. The water blowing machine, or trombe, as the French call it, consists of a reservoir of water AB, fig. 122, into the bottom of which the bent leaden pipe BCH is inserted; of a condensing vessel DE, into whose top the lower extremity H of the pipe is fixed, and of a pedestal P resting on the bottom of this vessel. When the water from the reservoir AB is descending through the part CH of the pipe, it is in contact with the external air by means of the orifices or tubes m, n, o, p; and by the principle of the lateral communication of motion in fluids, the air is dragged along with the water. This combination of air and water issuing from the aperture H, and impinging upon the surface of the stone pedestal P, is dispersed in various directions. The air being thus separated from the water, ascends into the upper part of the vessel, and rushes through the opening F, whence it is conveyed by the pipe FG to the fire at G, while the water falls to the lower part of the vessel, and is discharged by the openings M, N. That the greatest quantity of air may be driven into the vessel DE, the water should begin to fall at C with the least possible velocity; and the height of the lowest tubes above the extremity H of the pipe should be three-elevenths of the length of the vertical tube CH, in order that the air may move in the pipe FG with sufficient velocity.

405. A different form of the machine is shewn in fig. 123, where AB is the pipe of a conical form, and where the water which flows down AB is supplied with air by the pipes CB, DB.

406. In the machines of this kind used at Alvar in Dauphiny, the diameter of the conical pipe AB is 12 inches at A, and 5 at B, and only four tubes are used for admitting the air. This machine gives a powerful as well as an equal and continued blast, but the air is considered to be too moist and too cold.

407. Fabri and Dietrich imagined that the wind is produced by the decomposition of the water, or its transformation into gas, in consequence of the agitation and percussion of its parts. But M. Venturi, to whom we owe the first philosophical account of this machine, has shewn that this opinion is erroneous, and that the wind is supplied from the atmosphere, for no wind was generated when the lateral openings m, n, o, p were shut. The principal object, therefore, in the construction of water-blowing machines, is to combine as much air as possible with the descending current. For this purpose the water is often made to pass through a kind of culender placed in the open air, and perforated with a number of small triangular orifices. Through these apertures the water descends in many small streams; and by exposing a greater surface to the atmosphere, it carries along with it an immense quantity of air. The water is then conveyed to the pedestal P, fig. 122, by a pipe CH opened and enlarged at C, so as to be considerably wider than the end of the tube which holds the culender.

408. It has been generally supposed that the waterfall should be very high; but Dr Lewis has shewn, by a variety of experiments, that a fall of four or five feet is sufficient, 409. The rain wind is produced in the same way as the blast of air in water-blowing machines. When the drops of rain impinge upon the surface of the sea, the air which they drag along with them often produces a heavy squall, which is sufficiently strong to carry away the mast of a ship. The same phenomenon happens at land, when the clouds empty themselves in alternate showers. In this case, the wind proceeds from that quarter of the horizon where the shower is falling. The common method of accounting for the origin of the winds by local rarefaction of the air appears pregnant with insuperable difficulties; and there is reason to think that these agitations in our atmosphere ought rather to be referred to the principle which we have now been considering. The ventaroli which issue from volcanic mountains, arise from the air which is carried down the hollows by the falls of water. At the foot of the cascades which fall from the glazier of Roche Melon, Venturi found the force of the wind arising from the air dragged down by the water to be so strong, that it could scarcely be withstood. For farther information on this subject, the reader is referred to Lewis's Commerce of Arts, Wolfii Opera Mathematica, tom. i. p. 830, Journal des Mines, No. XCI, or Nicholson's Journal, vol. ii. 4to, p. 487, vol. xii. p. 48.

Sect. II. Bramah's Hydrostatic Press.

410. The machine invented by Mr. Bramah, depends upon the principle, that any pressure exerted upon a fluid mass is propagated equally in every direction. It is represented in fig. 124, where L is a strong metallic cylinder, furnished with a piston A, perfectly water-tight at the neck N. Into the side of this cylinder is inserted the end of the tube I, the interior orifice of which is closed by the valve at I. The other extremity of the tube communicates with the forcing pump DCH, by which water or other fluids may be driven into the cylinder L. The whole rests on a solid mass of masonry EF, or a firmly fixed wooden frame. A valve K, wrought by a screw, allows the water to return to G from the pipe M. The body to be crushed, broken, or pressed, is placed above the horizontal board B. Then, if any pressure is exerted on the surface of the water in the cylinder H, by means of the lever D, this pressure will be propagated to the cylinder L, and exert a certain force upon the piston A, varying with the respective areas of the sections of each cylinder. If the diameter of the cylinder H, is equal to the diameter of the cylinder L, and if a force of 10 pounds is exerted at the handle D, then the piston A will be elevated with a force of 10 pounds; if the diameter of H be one-half that of L, the piston A will be raised with a force of 40 pounds, because the area of the one piston is four times the area of the other. Or, in general, if D be the diameter of the cylinder L, d that of the cylinder H, and F the force exerted at the lever D, we shall have \( \frac{F \times D^2}{d^2} = \frac{F \times D^2}{d^2} \), which is the force exerted upon the piston B. Thus, if \( d = 2 \) inches, \( D = 24 \) inches, and \( F = 10 \) pounds, then \( \frac{F \times D^2}{d^2} = \frac{10 \times 24 \times 24}{2 \times 2} = 1440 \) pounds, the force with which the piston B is elevated. Now, as this force increases as \( d^2 \) diminishes, or as F and \( D^2 \) increase, there is no limit to the power of the engine; for the diameter of the cylinder L may be made of any size, and that of the cylinder H exceedingly small, while the power may be still farther augmented by lengthening the lever D. The same effects may be produced by injecting air into the pipe I by means of a large globe fixed at its extremity. Upon the same principles the power and motion of one machine may be communicated to another; for we have only to connect the two machines by means of a pipe filled with water, inserted at each extremity into a cylinder furnished with a piston. By this means the power which depresses one of the pistons will be transferred along the connecting pipe, and will elevate the other piston. In the same way water may be raised out of wells of any depth, and at any distance from the place where the power is applied; but we must refer the reader, for a detailed account of these applications, to the specification of the patent obtained by Mr. Bramah, or to Gregory's Mechanics, vol. ii. p. 120.

Sect. III. M. Manoury Dectot's Danaide.

411. This machine, invented by M. Manoury Dectot of Paris, consists of a cylindrical trough of tin-plate, nearly as high as it is broad, and having a hole in the centre of its bottom. It is fixed to a vertical axis of iron, which passes through the middle of the hole in the bottom, a vacant space being left all around to permit the water to escape. The axis turns with the trough upon a pivot, and is fixed above to a collar.

A drum of tin-plate, close above and below, is fixed upon the axis of the trough, and placed within the trough, so as to be concentric with it, and to leave only between the outer circumference of the drum and the inner circumference of the trough, an annular space not exceeding 1\(\frac{1}{2}\) inches. This annular space communicates with a space less than 1\(\frac{1}{2}\) inches, left between the bottom of the drum and the bottom of the trough, and divided into compartments by diaphragms fixed upon the bottom of the trough, and proceeding from the circumference to the central hole in the bottom of the trough.

The water comes from a reservoir above by one or two pipes, and makes its way into this annular space between the trough and drum. The bottom of these pipes corresponds with the level of the water in the trough, and they are directed horizontally, and as tangents to the mean circumference between that of the trough and of the drum. The velocity which the water has acquired by its fall along these pipes, makes the machine move round its axis, and this motion accelerates by degrees, till the velocity of the water in the space between the trough and drum equals that of the water from the reservoir; so that no sensible shock is perceived of the affluent water upon that which is contained in the machine.

This circular motion communicates to the water between the trough and drum a centrifugal force, in consequence of which it presses against the sides of the trough. This centrifugal force acts equally upon the water contained in the compartments at the bottom of the trough, but it acts less and less as this water approaches the centre.

The whole water then is animated by two opposite forces, viz. gravity, and the centrifugal force. The first tends to make the water run out at the hole at the bottom of the trough; the second, to drive the water from that hole.

To these two forces are joined a third, viz. friction, which acts here an important and singular part, as it promotes the efficacy of the machine, while in other machines it always diminishes that efficacy. Here, on the contrary, the effect would be nothing were it not for the friction, which acts as a tangent to the sides of the trough and drum.

By the combination of these three forces, there ought to result a more or less rapid flow from the hole at the bottom of the trough; and the less force the water has as it issues out, the more it will have employed in moving the machine, and of course in producing the useful effect for which it is destined.

The moving power is the weight of the water running in, multiplied by the height of the reservoir from which it flows above the bottom of the trough; and the useful effect is the same product diminished by half the force which the water retains when it issues out of the orifice below.

In order to ascertain, by direct experiment, the magnitude of this effect, MM. Prony and Carnot fixed a cord to the axis of the machine, which passing over a pulley, raised a weight by the motion of the machine. By this means, the effect was found to be $\frac{1}{5}$ of the power, and often approached $\frac{2}{5}$ without reckoning the friction of the pulleys, which has nothing to do with the machine. This effect exceeds that of the best overshot-wheels. See the Report of the Institute, 23rd August 1813; or Thomson's Annals of Philosophy, vol. ii. p. 412.

**Sect. IV. On Clepsydra or Water-Clocks.**

412. A clepsydra or water-clock, derived from κλέπτειν, to steal, and ὕδωρ, water, is a machine which measures time by the motion of water. The invention of this machine has been ascribed to Scipio Nasica, the cousin of Scipio Africanus, who flourished about two hundred years before the Christian era. It was well known, however, at an earlier period, among the Egyptians, who employed it to measure the course of the sun. It is highly probable that Scipio Nasica had only the merit of introducing it into his native country. These machines were in use for a very long period, and continued to be employed as measurers of time till the invention of the pendulum clock enriched the arts and sciences.

413. The earliest, and probably the simplest water-clock, consisted of a hollow cone $A$ (fig. 125), perforated at its apex, and of a solid cone $B$, which exactly fitted the interior of $A$. The aperture of the cone $A$ was of such a size in relation to the contents of $A$, that it discharged all the water in $A$ in the course of the shortest day in winter. The length of the cone was divided into twelve parts, which indicated the hours as the water was discharged. As the day became longer, the line in which the water was discharged required to be increased; and, in order to effect this, the solid cone $B$ was introduced into the hollow cone $A$, and, in proportion to the depth of its immersion, the water flowed with less facility, and more slowly, from the aperture at the vertex of the cone $A$. By this means the time of emptying the cone was accommodated to the varying length of the day, and the adjustment for this purpose was made by a graduated scale $BC$, upon the handle of the solid cone.

413. The clepsydra, invented by Ctesibius of Alexandria, was an interesting machine. The water, which indicated the sydria progress of time by the gradual descent of its surface, flowed in the form of tears from the eyes of the human figure. Its head was bent down with age: its look was dejected, while it seemed to pay the last tribute of regret to the fleeting moments as they passed. The water which was thus discharged was collected in a vertical reservoir, where it raised another figure, holding in its hand a rod, which, by its gradual ascent, pointed out the hours upon a vertical column. The same fluid was afterwards employed in the interior of the pedestal, as the impelling power of a piece of machinery, which made this column revolve round its axis in a year, so that the months and the days were always shewn by this index, whose extremity described a vertical line divided according to the relative lengths of the hours of day and night. Among the ancients the length of the hours varied every day, and even the hours of the day differed in length from those of the night; for the length of the day, or the interval between sunrise and sunset, was always divided into twelve equal parts, while the length of the night, or the interval between sunset and sunrise, was divided into the same number of parts, for hours. A further description of this beautiful machine, and others of the same nature, may be seen in Perrault's Vitruvius.

415. The method of constructing clepsydra, when the vessel from which the fluid issues is cylindrical or of any other form, has been shewn in Prop. VII. Part II. Instead of dividing the sides of the vessel, for a scale to ascertain the descent of the fluid surface, the following method may be adopted. In the bottom of the cylindrical vessel $ABCD$ (fig. 126.), which is about twelve inches high, and four inches in diameter, is inserted a small glass adjuage $E$, which discharges the water in the vessel by successive drops. A hole $F$, about half an inch in diameter, is perforated in the cover $AB$, so as to allow the glass tube $GI$, about sixteen inches long, and half an inch in diameter, to move up and down without experiencing any resistance. To the extremity of this tube is attached the ball $I$, which floats on the surface of the water in the vessel, and is kept steady, either by introducing a quantity of mercury into its cavity, if it be hollow, or by suspending a weight, if it is a solid which does not sink in water. When the vessel is filled with water, the ball $I$ will be at the top $AB$; then, in order to graduate the tube $C$, canal ee, supplied with a constant and equal stream by the syphon d, has at each end f1, f2, open pipes f1, f2, of exactly equal bores, which deliver the water that runs along the canal e, alternately into the vessels g1, g2, in such a quantity as to raise the water from the mouth of the tantalus t, exactly in an hour. The canal ee is equally poised by the two pipes f1, f2, upon a centre r; the ends of the canal e are raised alternately, as the cups zz are depressed, to which they are connected by the lines running over the pulleys II. The cups zz are fixed at each end of the balance mm, which moves up and down upon its centre v. n1, n2, are the edges of two wheels or pulleys, moving different ways alternately, and fitted to the cylinder o by oblique teeth, both in the cavity of the wheel and upon the cylinder, which, when the wheel n moves one way, that is, in the direction of the minute-hand, meet the teeth of the cylinder, and carry the cylinder along with it, and slip over those of the cylinder when n moves the contrary way, the teeth not meeting, but receding from each other. One or other of these wheels nn continually moves o in the same direction, with an equable and uninterrupted motion. A fine chain goes twice round each wheel, having at one end a weight X, always out of the water, which equiperates with y at the other end, when kept floating on the surface of the fluid in the vessel q, which y must always be; the two cups z, z, one at each end of the balance, keep it in equilibrium, till one of them is forced down by the weight and impulse of the water, which it receives from the tantalus ti. Each of these cups z, z, has likewise a tantalus of its own h, h, which empties it after the water has run from q, and leaves the two cups again in equilibrium: q is a drain to carry off the water. The dial-plate, &c., needs no description. The motion of the clepsydra is effected thus: As the end of the canal ee, fixed to the pipe f1, viz. the lowest in the figure, all the water supplied by the syphon runs through the pipe f1, into the vessel g1, till it runs over the top of the tantalus t; when it immediately runs out at i, into the cup Z, at the end of the balance m, and forces it down; the balance moving on its centre V. When one side of m is brought down, the string which connects it to f1, running over the pulley l, raises the end f1, of the canal e, which turns upon its centre r, higher than f2; consequently, all the water which runs through the syphon d, passes through f2 into g2, till the same operation is performed in that vessel, and so on alternately. As the height to which the water rises in g in an hour, viz. from S to t, is equal to the circumference of n, the float y rising through that height along with the water, allows the weight X to act upon the pulley n, which carries with it the cylinder o; and this, making a revolution, causes the index k to describe an hour on the dial-plate. This revolution is performed by the pulley n1; the next is performed by n2, whilst n1 goes back, as the water in g1 runs out through the tantalus; for y must follow the water, as its weight increases out of it. The axis o always keeps moving the same way; the index p describes the minutes; each tantalus must be wider than the syphon, that the vessels gg may be emptied as low as S, before the water returns to them.

360. For farther information respecting subjects connected with hydrodynamics, see Mechanics, Resistance of Fluids, Water-Works, &c. (N. N. N.) HYDROGRAPHICAL Charts or Maps, usually called sea-charts, are projections of some part of the sea or coast, for the use of navigation. In these are laid down all the rhumbs or points of the compass, the meridians, parallels, &c. with the coasts, capes, islands, rocks, shoals, shallows, &c. in their proper places and proportions.