a small island in the Pacific Ocean, near the east coast of the island of Luzon. Long. 124. E. Lat. 14. 10. N.

**TIDES.**

The alternate rising and falling which are observed to take place in the surface of the waters, generally twice in the course of a lunar day, or of 24h. 49m. of mean solar time, on most of the shores of the ocean, and in the greater part of bays, firths, and rivers which communicate freely with it, are the phenomena denominated the tides. These form what are called a flood and an ebb, a high and low water. The whole interval between high and low water is called a tide; the water is said to flow and to ebb; and the rising is called the flood tide, and the falling the ebb tide. This rise or fall of the waters is exceedingly different at different places, and is also variable everywhere. At Plymouth, for instance, it is sometimes twenty feet between the greatest and least heights of the water in one day, and sometimes only twelve feet. These different heights of tide succeed each other in a regular series, diminishing from the greatest to the least, and then increasing from the least to the greatest. The greatest is called a spring tide, and the least a neap tide. This series is completed in about fifteen days. More careful observation shows that two series are completed in the time of a lunation. For the spring tide in any place happens at a certain interval of time, generally between one and two days, after new or full moon; and the neap tide at a certain interval after the quarter; or, more accurately speaking, the spring tide occurs when the moon has got a certain number of degrees eastward either from the line of conjunction with the sun or of opposition to him, and the neap tide when she is at a certain distance from her first or last quadrature. Thus the whole series of tides appears to be chiefly regulated by the moon, and seems to be only to a small extent under the influence of the sun; for it is further observed that high water happens when the moon has nearly a certain position with respect to the meridian of the place of observation, preceding or following the moon's southing by a certain interval of time; which, at new and full moon, is generally not far from being constant with respect to the same place, but very different in different places; whereas the interval between the time of high water and noon changes almost everywhere about six hours in the course of a fortnight.

The interval between two succeeding high waters is variable. It is shortest about new and full moon, being then about 12h. 19m.; and about the time of the moon's quadratures it is 12h. 30m. But these intervals are somewhat different at different places. The tides in similar circumstances are greatest when the moon is in the equator, and at her smallest distance from the earth, or in her perigee; and, gradually diminishing, are smallest when she is in her apogee, and farthest from the equator.

Such are some of the more general and regular phenomena of the tides. In certain places there are four tides in the lunar day, in others but one; and in some there is scarcely any perceptible variation of level, which regularly keeps time with the moon. The tides being important to all commercial nations, great exertions have recently been made to obtain the means of predicting them. Some account

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1 Dr Grainger published a professional work, which has escaped the notice of some of his biographers: "Historia Febris Anomale Batavae, annorum 1746, 1747, 1748, &c. Accedunt Monita Siphylica. Auctore Jacobo Grainger, M.D." Edinb. 1753, 8vo. The latter tract is a reprint of his inaugural dissertation, on taking his degree. An edition of this volume was subsequently published in Holland. of these and other empirical researches, which have of late been conducted on an extensive scale and with great care, will be given in the latter part of this article.

It is of tides, most probably, the Bible speaks, when God is said to set bounds to the sea, and to say, "thus far shall it go, and no farther." Homer would be the earliest profane author who notices the tides, if indeed it be to them he refers (in the 12th book of the Odyssey) when he speaks of Charybdis rising and retiring thrice a day. Herodotus and Diodorus Siculus speak more distinctly of the tides in the Red Sea. Pytheas of Marseille is the first who says anything of their cause. According to Strabo, he had been in Britain, where he must have observed the tides of the ocean. Plutarch says expressly that Pytheas ascribed them to the moon. It is remarkable that Aristotle says so little about the tides. The army of Alexander, his pupil, were startled at first seeing them near the Persian Gulf; and Aristotle would probably be well informed of all that had been observed there. But in all his writings there are only three passages concerning them, and these are very trivial. In one place he speaks of great tides in the north of Europe; in another, of their having been ascribed by some to the moon; and in a third, he says, the tide in a great sea exceeds that in a small one.

The Greeks had little opportunity of observing the tides. The conquests and the commerce of the Romans gave them more acquaintance with them. Caesar speaks of them in the fourth book of his Gallic War. Strabo, after Posidonius, classes the phenomena into daily, monthly, and annual. He observes, that the sea rises as the moon approaches the meridian, whether above or below the horizon, and falls again as she rises or falls; that the tides increase at the time of new and full moon, and are greatest at the summer solstice. Pliny explains the phenomena at some length, and ascribes them to the sun and moon dragging the waters along with them (b. ii. c. 97). Seneca (Nat. Quest. iii. 28) speaks of the tides with correctness; and Macrobius (Sonn. Scip. i. 6) gives a tolerable description of their motions. Such phenomena naturally exercise human curiosity as to their cause. Plutarch (Placit. Phil. iii. 17), Galileo (Syst. Mund. dial. 4), Riccioli in his Almagest, ii. p. 374, and Gassendi, ii. p. 27, have collected most of the notions of their predecessors on the subject; but they are of so little importance as not to deserve our notice. Kepler, in accounting for the tides (De Stella Maris, and Epit. Astron. p. 555), had evidently been aware of the principle of gravitation, but not of the law. He says that all bodies attract each other, and that the waters of the ocean would all go to the moon were they not retained by the attraction of the earth. He then proceeds to explain their elevation under the moon and on the opposite side, because the earth is less attracted by the moon than the nearer waters, but more than the waters which are more remote. The honour of a complete explanation of the tides in a general way was reserved for Sir Isaac Newton. He laid hold of this class of phenomena as the most incontestable proof of universal gravitation, and has given a most beautiful and symptical view of the whole subject; contenting himself, however, with merely exhibiting the chief consequences of the general principle, and applying it to the phenomena with singular address.

The investigation of the phenomena of the tides has been justly considered as uniting some of the greatest difficulties that occur in the various departments of natural philosophy and astronomy. It implies, first, a knowledge of the laws of gravitation, concerned in the determination of the forces immediately acting on the sea, and of the periods and distances of the celestial bodies, which modify the magnitudes and combinations of these forces; and, secondly, of the hydraulic theories of the resistances of fluids, and of the motions of waves and undulations of all kinds, and of the theoretical determination of the form and density of the earth, as well as of the geographical observation of the breadth and depth of the seas and lakes which occupy a part of its surface; so that the whole subject affords abundant scope for the exercise of mathematical skill, and still more for the employment of that invention and contrivance which enables its possessor to supersede the necessity of prolix computations wherever they can be avoided.

The history of the theory of the tides is naturally divided into several periods in which its different departments have been progressively cultivated. The ancients from the times of Posidonius and Pytheas, and the moderns before Newton, were contented with observing the general dependence of the tides on the moon, as following her transit at an interval of about two hours, and their alternate increase and decrease not only every fortnight, but also in the lunar period of about eight years. The second step consisted in the determination of the magnitude and direction of the solar and lunar forces, by which the general effects of the tides were shown, in the Principia, to be the necessary consequences of these forces. The third great point was the demonstration of Maclaurin, that the form of an elliptic spheroid affords an equilibrium under the action of the disturbing forces concerned; while the further contemporary illustrations of the subject by Euler and Bernoulli, though they afforded some useful details, involved no new principle that can be put in competition with Maclaurin's demonstration. The fourth important step was made by Laplace, who separated the consideration of the form affording mere equilibrium, from that of the motion occasioned by the continual change of that form; while former theorists had taken it for granted that the surface of the sea very speedily assumed the figure of a fluid actuated by similar forces, but remaining perfectly at rest, or assuming instantly the form in question. Laplace's computation is however limited to the case of an imaginary ocean, of a certain variable depth, assumed for the convenience of calculation, rather than for any other reason. Dr Thomas Young has extended Laplace's mode of considering the phenomena to the more general case of an ocean covering a part only of the earth's surface, and more or less irregular in its form; he has also attempted to comprehend in his calculations the precise effects of hydraulic friction on the times and magnitudes of the tides. As far as the resistance may be supposed to vary in the simple ratio of the velocity, Dr Young's theory is sufficiently complete, and explains several of the peculiarities which are otherwise paradoxical in their appearance; but there still remains a difficulty to be combated with respect to the effects of a resistance proportional to the square of the velocity, and this, it is hoped, will be in great measure removed in the present article, which, however, from the space that is allotted to it, must be considered rather as a supplementary fragment than as a complete treatise. This theory will be divided into four sections: the first relating to the contemporaneous progress of the tides through the different seas and oceans, as collected from observation only; the second to the magnitude of the disturbing forces tending to change the form of the surface of the earth and sea; the third to the theory of compound vibrations with resistance; and the fourth to the application of this theory to the progress and successive magnitudes of the tides, as observable at any one port.

Sect. I.—Of the Progress of Contemporary Tides, as inferred from the times of High Water in different Ports.

The least theoretical consideration relating to the tides, is that of their progress through the different parts of the ocean, and of its dependent seas. The analysis of these ought to be very completely attainable from direct obser- vation, if the time of high water had been accurately observed at a sufficient number of ports throughout the world; and, on the other hand, if the earth were covered in all parts with a fluid of great and nearly uniform depth, the tides of this fluid would be so regular, that a very few observations would be sufficient to enable us to deduce the whole of the phenomena from theory, and to trace the great waves, which would follow the sun and moon round the globe, so as to make its circuit in a day, without any material deviation from uniformity of motion and succession.

Having collected, for the actual state of the sea and continents, an abundant store of accurate observations of the precise time of high water with regard to the sun and moon, for every part of the surface, and having arranged them in a table according to the order of their occurrence, as expressed in the time of any one meridian, we might then suppose lines to be drawn on a terrestrial globe, through all the places of observation, in the same order; and these lines would indicate, supposing the places to be sufficiently numerous, so as to furnish a series of tides very nearly contemporary, the directions of the great waves, to which that of the progress of the tides in succession must be perpendicular.

If, however, we actually make such an attempt, we shall soon find how utterly inadequate the observations that have been recorded are, for the purpose of tracing the forms of the lines of contemporary high water with accuracy or with certainty, although they are abundantly sufficient to show the impossibility of deducing the time of high water at any given place from the Newtonian hypothesis, or even that of Laplace, without some direct observation. It might at least be supposed very easy to enumerate the existing observations, scanty as they may be, in a correct order; but there is a number of instances in which it is wholly uncertain whether the time observed at a given port relates to the tide of the same morning in the open ocean, or to that of the preceding evening. This inconvenience may, however, in some measure be remedied, by inserting such places in two different parts of the table, at the distance of twelve and a half hours from each other. The following table is the result of an approximation obtained in this manner, the principal hour-lines having been partially traced on a map of the world, in order to afford some little direction to the correct insertion of the times of high water without the material error of half a day.

### Time of High Water at the Full and Change of the Moon, reduced to the Meridian of Greenwich.

| Longitude | H.W.Gr.T. | Longitude | H.W.Gr.T. | Longitude | H.W.Gr.T. | |-----------|-----------|-----------|-----------|-----------|-----------| | S. Georgia | 2.26W. | Brest | 9.18 | Socotra C. | 20.30W. | | Cape Good Hope | 1.14E. | Bayonne | 0.8 | Guardafui | 21.30 | | St Helena | 0.23W. | C. St Vincent | 0.36 | Ulletta | 10.6 | | Cape Corse | 0.7 | Corum | 0.37 | Huahine | 10.4 | | Rio Janeiro | 2.53 | Belleisle | 0.12W. | Shoreham | 0.1 | | L. Martin Vaz | 1.66 | Palmiras Pt. | 18.12 | Foul Pt. Mad. | 20.41 | | L. Ascension | 0.57 | Port Cornwallis | 17.49 | Botany Bay | 13.55 | | Christmas Sound | 4.46 | Rochelle | 0.5 | St Valery en Caux | 0.3E. | | St Jago | 1.34 | Vannes | 0.11 | Macao | 16.26W. | | Port Desiré | 4.20 | St Paul de Leon | 0.16 | St Valery sur Somme | 0.6E. | | St Helena, S.A. | 4.40 | Morlaix | 0.16 | Dunmore | 0.5W. | | Quibo | 5.29 | Rochfort | 0.4 | Brighton | 0.1 | | Sierra Leone | 0.53 | Bear Island | 5.20 | Dublin | 0.25W. | | Easter I. | 7.19 | Christmas Island | 19.24 | Abbeville | 0.7E. | | St Julian's | 4.35 | Chloe | 5.0 | Beachy Head | 0.1E. | | Maragogi Mouth | 3.20 | Cape Clear | 0.35 | Cowes | 0.6W. | | St John's, Newf. | 3.30 | Annamocka | 11.39 | Needles | 0.6W. | | Guadeloupe | 4.7 | St Peter and Paul | 13.25 | Anhalt | 0.47E. | | Panama | 5.21 | Awatia | 12.5 | Boulogne | 0.6 | | Tortugas | 4.51 | Kinself | 0.34 | Hastings | 0.3 | | Cape Blanco | 6.16 | Eddystone | 0.18 | Deal Castle | 0.6 | | Bermuda | 4.14 | Falmouth | 0.20 | Dover | 0.5 | | Martinique | 4.5 | Rotterdam I. | 12.21 | Dungeness | 0.4 | | Guyanagul | 5.17 | Drake's I. Plymouth | 17.52 | Dieppe | 0.4 | | Senegal | 1.6 | Plymouth | 0.17 | Eagle L. | 21.26W. | | Callao | 4.8 | Avranches | 0.3 | Almirantes | 23.48 | | Halifax | 4.14 | Esmee | 11.25 | Curreuse | 21.17 | | Marquesas | 9.16 to 11.46 | St Maloes | 0.8 | Portsmouth | 23.36 | | Quebec | 4.44 | London | 12.29 | Ostend | 0.13E. | | Cape La Hogue | 0.8 | Tonga Taboo | 12.29 | Niaspon | 0.1 | | Gibraltar | 0.29 | Granville | 0.6 | Great Yarmouth | 0.8E. | | Tahoga, Pan. Bay | 5.22 | Pudya | 13.1 | Alderney | 0.9W. | | Funchal | 1.8 | St Francisco | 8.3 | Bergen | 0.21E. | | Portobello | 5.19 | Cork | 0.34 | False Bay | 22.45W. | | Cape Bejador | 0.58 | Bristol | 0.10 | Drotheim | 0.41E. | | Churchill R. | 6.17 | Barfleur | 0.5 | Roanen | 0.4W. | | P. of Wales Fort | 13.37 | Cherbourg | 0.6 | Aberdeen | 0.9 | | Terceira | 1.49 | Venus Pt. Otals | 2.55 | North Cape | 1.43E. | | New York | 4.57 | Mauritius | 20.10 | Leith and Edinburgh | 0.13W. | | Cape Henlopen, Ving | 5.1 | Lizard | 0.21 | Amsterdam | 0.19E. | | Caliz | 0.25 | Nooks Sound | 8.27 | Rotterdam | 0.18 | | Karakakoon Bay | 10.24 | Guernsey | 0.9 | London Bridge | 0.0 | | Virgin Cape, Pat. | 4.32 | Pulo Condore | 18.53 | Archangel | 2.36E. | | Valparaiso | 4.49 | Calcutta | 16.6 | Bordeaux | 0.2W. | | Cape Charles | 4.57 | Seychelles Alan | 20.16 | Hamburg | 0.40E. | | Goree Island | 1.10 | Stromness | 0.14 | Bremen | 0.35E. | | York Port | 6.9 | N.Zealand Q.C.S. | 12.13 | Antwerp | 0.18E. | | Lisboa | 0.37 | Honfleur | 0.1E. | Scott Head | 0.3 | | Nantes, Roi | 0.6 | Havre | 0.0 | Lynn | 0.2 | | Tana | 11.19 | Caen | 0.1 | Hague | 0.17 | | | | | | Leostoffe | 0.7 | | | | | | London Bridge | 0.0 |

(39.45) It may be immediately inferred from this table, first, that the line of contemporary tides is seldom in the exact direction of the meridian, as it is supposed to be universally in the theory of Newton and of Laplace; except perhaps, the line for the twenty-first hour in the Indian Ocean, which appears to extend from Socotra to the Almirantes and the Isle of Bourbon, lying nearly in the same longitude. Secondly, the southern extremity of the line advances as it passes the Cape of Good Hope, so that it turns up towards the Atlantic, which it enters obliquely, so as to arrive, nearly at the same moment, at the Island of Ascension, and at the Island of Martin Vaz, or of the Trinity. Thirdly, after several irregularities about the Cape Verde Islands, and in the West Indies, the line appears to run nearly east and west from St Domingo to Cape Blanco, the tides proceeding due northwards; and then, turning still more to the right, the line seems to run north-west and south-east, till at last the tide runs almost due east up the British Channel and round the north of Scotland into the Northern Ocean, sending off a branch down the North Sea to meet the succeeding tide at the mouth of the Thames. Fourthly, towards Cape Horn, again, there is a good deal of irregularity; the hour-lines are much compressed between South Georgia and Tierra del Fuego, perhaps on account of the shallower water about the Falkland Islands and South Shetland. Fifthly, at the entrance of the Pacific Ocean, the tides seem to advance very rapidly to New Zealand and Easter Island; but here it appears to be uncertain whether the line of contemporary tide should be drawn nearly north and south from the Galapagos to Tierra del Fuego, or north-east and south-west from Easter Island to New Zealand; or whether both these partial directions are correct: but on each side of this line there are great irregularities, and many more observations are wanting before the progress of the tide can be traced with any tolerable accuracy, among the multitudinous islands of the Pacific Ocean, where it might have been hoped that the phenomena would have been observed in their greatest simplicity, and in their most genuine form. Lastly, of the Indian Ocean the northern parts exhibit great irregularities, and among the rest they afford the singular phenomenon of one tide in the day, observed by Halley in the port of Tonkin, and explained by Newton in the Principia: the southern parts are only remarkable for having the hour-lines of contemporary tides considerably crowded between New Holland and the Cape of Good Hope, as if the seas of these parts were shallower than elsewhere.

These inferences respecting the progress of the tides are not advanced as the result of any particular theory, nor even as the only ones that might possibly be deduced from the table. Thus the supposition that the direction in which the tides advance must be perpendicular to the hour-lines of contemporary tides, is not by any means absolutely without exception, since a quadrangular lake, with steep shores in the direction of the meridian, would have the times of high water the same for every point of its eastern or western halves respectively, and there could be no correctly defined direction of the hour-lines in such a case. But if any portion of the sea could be considered as constituting such a lake, its properties would be detected by a sufficient number of observations of high water; and the existing table does not appear to indicate any such cases that require to be otherwise distinguished than as partial irregularities. There may also be some doubt respecting the propriety of the addition of twelve and a half hours that has been made to the time of high water in the north-eastern parts of the Atlantic: but it seems extremely improbable that the same tide should travel north-easterly into the English Channel and into the Northern Ocean, and at the same time westerly across the Atlantic, as it must be supposed to do, if it were considered as primarily originating in the neighbourhood of the Bay of Biscay. On the other hand, the bending of the great wave round the continents of Africa and Europe seems to be very like the sort of refraction which takes place on every shelving coast with respect to the common waves, which, whatever may have been their primitive origin, acquire always, as they spread, a direction more and more nearly parallel to that of the coast which they are approaching: and the suppositions which have been here advanced respecting the succession of the tides in different ports, allowing for the effect of a multitude of irregularities proceeding from partial causes, appear to be by far the most probable that can be immediately inferred from the table, at least in its present state of imperfection.

Sect. II.—Of the Disturbing Forces that occasion the Tides.

Since the phenomena of the tides, with regard to their progress through the different oceans and seas, as they exist in the actual state of the earth's surface, appear to be too complicated to allow us to hope to reduce them to computation by means of any general theory, we must, in the next place, confine our attention to the order in which the successive changes occur in any single port; and having determined the exact magnitude of the forces that tend to change the form of the surface of the ocean at different periods, and having also examined the nature of the vibratory motions of which the sea, or any given portion of it, would be susceptible, in the simplest cases, after the cessation of the disturbing forces, we must afterwards endeavour to combine these causes, so as to adapt the result to the successive phenomena which are observed at different times in any one port.

Theorem A. ("E."—Nicholson's Journal, July 1813.) The disturbing force of a distant attractive body, urging a particle of a fluid in the direction of the surface of a sphere, varies as the sine of twice the altitude of the body.

The mean attraction exerted by the sun and moon on all the separate particles composing the earth, is exactly compensated by the centrifugal force derived from the earth's annual revolution round the sun, and from its monthly revolution round the common centre of gravity of the earth and moon; but the difference of the attractions exerted at different points of the earth, must necessarily produce a disturbing force, depending on the angular position of the point with regard to the sun or moon, since the centrifugal force is the same for them all; the disturbing force being constantly variable for any one point, and depending partly on the difference of the distance of the point from the mean distance, and partly on the difference of the direction of the luminary from its direction with respect to the centre, or, in other words, on its parallax.

In the case of a sphere covered with a fluid, it will be most convenient for computation to consider both these forces as referred to the direction of the circumference of the sphere, which will differ but little from that of the fluid; and it will appear that both of them, when reduced to this direction, will vary as the product of the sine and cosine of the distance from the diameter pointing to the luminary, that is, as half the sine of twice the altitude; for the difference of gravitation, which depends on the difference of the distance, will always vary as the sine of the distance from the bisecting plane perpendicular to that diameter, and will be reduced to the direction of the surface by diminishing it in the ratio of the cosine to the radius; and the effect of the difference of direction will be originally proportional to the sine of the distance from the diameter, and will in like manner be expressed, when reduced, by the product of the sine and cosine; and each force, thus reduced, will be equal, where it is greatest, to half of its primitive magnitude, since $\sin \cos 45^\circ = \frac{1}{2}$. Thus, the gravitation towards the moon at the earth's surface, is to the gravitation towards the earth as 1 to 70 times the square of 60½, or to 256 217: and the former disturbing force is to the whole of this as 2 to 60 at the point nearest the moon, and the second as 1 to 60½ at the equatorial plane, and the sum of both, re- duced to the direction of the circumference, where greatest, as 3 to 121, that is, to the whole force of the earth's gra- vitation, as 1 to 10 334 000; and in a similar manner we find, that the whole disturbing force of the sun is to the weight of the particles as 1 to 25 736 000." Or, if we call the moon's horizontal parallax \( p \), and substitute \( \frac{1}{p} \) for the distance, the whole of the lunar disturbing force in the di- rection of the surface will be \( \frac{3}{2} \cdot \frac{p^2}{70} = \frac{3}{140} p^2 \); or, if \( z \) be the moon's zenith distance from any point of the surface, \( f = \frac{3}{70} p^2 \sin \cos z \).

**Theorem B.** [F.] The inclination of the surface of an oblong spheroid, slightly elliptical, to that of the inscribed sphere, varies as the sine of twice the distance from the circle of contact; and a particle resting on any part of it, without friction, may be held in equilibrium by the at- traction of a distant body [situated in the direction of the axis].

If a sphere be inscribed in an oblong spheroid, the ele- vation of the spheroid above the sphere must obviously be proportional, when measured in a direction parallel to the axis of the spheroid, to the ordinate of the sphere, that is, to the sine of the distance from its equator; and when re- duced to a direction perpendicular to the surface of the sphere, it must be proportional to the square of that sine; and the tangent of the inclination to the surface of the sphere, which is equal to the fluxion of the elevation di- vided by that of the circumference, must be expressed by twice the continual product of the sine, the cosine, and the ellipticity, or rather the greater elevation \( e \), the radius be- ing considered as unity: so that the elevation \( e \) will also express the tangent of the inclination where it is greatest, since \( 2 \sin \cos 45° = 1 \); and the inclination will be every- where as the product of the sine and cosine.

If, therefore, the density of the elevated parts be con- sidered as evanescent, and their attraction be neglected, there will be an equilibrium, when the ellipticity is to the radius as the disturbing force to the whole force of gravi- tation; for each particle situated on the surface will be actuated by a disturbing force tending towards the pole of the spheroid, precisely equal and contrary to that portion of the force of gravitation which urges it in the opposite di- rection down the inclined surface. Hence, if the density of the sea were supposed inconsiderable in comparison with that of the earth, the radius being 20 889 000 feet, the greatest height of a lunar tide in equilibrium would be 2'0166 feet, and that of a solar tide 8'097; that is, suppos- ing the moon's horizontal parallax about 57', and her mass \( \frac{1}{8} \) of that of the earth.

**Theorem C.** [Gr.] The disturbing attraction of the thin shell contained between a spheroidal surface and its in- scribed sphere, varies in the same proportion as the incli- nation of the surface, and is to the relative force of gravity depending on that inclination, as three times the density of the shell to five times that of the sphere.

We may imagine the surface of the spheres to be divided by an infinite number of parallel and equidistant circles, beginning from any point at which a gravitating particle is situated, and we may suppose all these circles to be divided by a plane perpendicular to the meridian of the point, and consequently bisecting the equatorial plane of the spheroid: it is obvious, that if the elevations on the opposite sides of the plane be equal at the corresponding points of each circle, no lateral force will be produced; but when they are unequal, the excess of the elevated matter on one side above that of the other side will produce a disturbing or lateral force. Now, the elevation being everywhere as the square of the distance \( x \) from the equatorial plane, we may call it \( e x^2 \), and the difference corresponding to any point of that semicircle which is the nearer to the pole of the spher- oid, will be \( e (x^2 - x'^2) = e (x^2 + x'^2) (x^2 - x'^2) \). But \( x^2 + x'^2 \) is always twice the distance of the centre of the supposed circle from the equatorial plane; and the distance of this centre from that of the sphere will be \( \cos \phi \), if \( \phi \) be the angular distance of the circle from its pole; and, calling \( \rho \) the distance of this pole from the equatorial plane of the spheroid, the distance in question will be \( \cos \phi \sin \phi \), and \( x^2 + x'^2 = 2 \cos \phi \sin \phi \); and the difference \( x^2 - x'^2 \) is twice the actual sine of the arc \( A \) in the supposed circle, that is, twice the natural sine, reduced in the ratio of unit to the radius of this circle, which is \( \sin \phi \), reduced again to a direction perpendicular to the equatorial plane; whence \( x^2 - x'^2 = 2 \sin \phi \cos \phi \); and \( x^2 - x'^2 = 4 \sin \phi \cos \phi \). Hence it follows, that, in different positions of the gravitating particle, the effective elevation at each point of the surface, similarly situated with respect to it, is as the product of the sine and cosine of its angular distance \( \phi \) from the equatorial plane, the other quantities concerned remaining the same in all positions. But the inclination of the surface of the spheroid, as well as the original disturbing force, varies in the same proportion of the product of the sine and cosine of the distance \( \phi \); consequently the sum of this disturbing attraction and the ori- ginal force will also vary as the inclination of the surface, and may be in equilibrium with the tendency to descend towards the centre, provided that the ellipticity be duly commensurate to the density of the elevated parts.

Now, in order to find the actual magnitude of the dis- turbing attraction for a shell of given density, we must com- pute the fluent of \( 4 e \sin \phi \cos \phi \sin \phi \cos \phi \), re- duced first according to the distance and direction of each particle from the given gravitating particle; and we must compare the fluent with \( \frac{4}{3} \pi \), the attraction of the whole sphere at the distance of the radius, or unity. But for the angle \( \phi \), the portion of the force acting in the common di- rection of \( \sin \phi \) is to the whole attraction at the same dis- tance as \( \sin \phi \) to 1, so that the attractive force of any point of the semicircle will be \( 4 e \sin \phi \cos \phi \sin \phi \cos \phi \), and its fluxion will be as \( \sin^2 \phi \cos \phi \), of which the fluent is \( \frac{1}{2} \phi - \frac{1}{2} \phi \cos \phi \), or when \( \phi = 180° \), \( \frac{1}{2} \pi \), and \( \frac{1}{2} \phi \sin \phi \) will express the effect of the disturbing attraction of the semi- circles, of which \( \phi \) is the radius, reduced to the direc- tion of the middle point, of which the distance is \( 2 \sin \frac{1}{2} \phi \); the reduction for this distance is as its square to 1; and for the direction, as the distance to \( \sin \phi \), together making the ratio of \( \frac{\sin \phi}{8 \sin^2 \phi} \), and the ultimate fluxion of the force will be \( 2e \sin \phi \cos \phi \sin \phi \cos \phi \sin \phi \cos \phi \). \( d\phi = 2e \sin^2 \phi \cos^2 \phi \sin \phi \cos \phi \sin \phi \cos \phi \); but \( \sin \phi = 2 \sin \phi \cos \phi \), and the fraction becomes \( \frac{8 \sin^2 \phi \cos^2 \phi}{8 \sin^2 \phi} \cos \phi = \cos^3 \phi \cos \phi = \cos^3 \phi \cos \phi - \cos^3 \phi \cos \phi = 2 \cos^3 \phi \cos \phi - \cos^3 \phi \cos \phi \). Now, tak- ing the fluent from \( \phi = 0 \) to \( \phi = 180° \), we have \( 2 \int \cos^3 \phi \cos \phi \times 2d\phi = \frac{4}{3} \), the differ- ence being \( \frac{3}{5} \cdot \frac{4}{3} = \frac{4}{5} \), whence the fluent of the force is found

\[ 2\pi \sin \cos \phi \times \frac{4}{5} \times \frac{1}{n} \] calling the density of the fluid \( \frac{1}{n} \)

or, where it is greatest, \( \sin \cos \phi \) being \( \frac{1}{3}, \frac{4}{5} \), while

the attraction of the sphere itself is \( \frac{4}{3} \pi \), which is to \( \frac{4}{5} \pi \) as

\( \frac{3e}{5n} \); and since the elevation \( e \) expresses also the maximum of the relative force of gravity depending on the tangent of the inclination (Theorem B), it is obvious that the disturbing attraction \( \frac{3e}{5n} \) must be to the relative force \( e \) as

\( \frac{3}{n} \) to 5.

Corollary 1. If \( n = 1 \), as in a homogeneous fluid sphere or spheroid, the disturbing attraction becomes \( \frac{3}{5} e \), and this attraction, together with the primitive force \( f \), must express the actual elevation \( e \), or \( \frac{3}{5} e + f = e \), whence \( f = \frac{2}{5} e \), and \( e = \frac{5}{2} f \), giving 2-024 and 5-042 for the magnitude of the solar and lunar tides, when \( f = .8097 \) and 2-0166 respectively. But this is obviously far from the actual state of the problem.

Corollary 2. Supposing \( n = 54 \) (see Quarterly Journal of Science, April 1820), we have \( \frac{3e}{27} + f = e \), and \( e = \frac{27}{24} f = \frac{9}{8} f \); so that the height of the primitive tides of an ocean of water, covering the whole surface of the earth, such as it actually is, ought to be .911 for the solar, and 2-27 for the lunar disturbing force; that is, supposing the sea without inertia, so as to accommodate itself at once to the form of equilibrium. But, in the actual state of the irregularities of the seas and continents, it is impossible to pay any regard to this secondary force, since the phenomena do not justify us in supposing the general form of the surface of the ocean such as to give rise to it.

Theorem D. [H.] When the horizontal surface of a liquid is elevated or depressed a little at a given point, the effect will be propagated in the manner of a wave, with a velocity equal to that of a heavy body which has fallen through a space equal to half the depth of the fluid, the form of the wave remaining similar to that of the original elevation or depression. Dr Young's Elementary Illustrations of the Celestial Mechanics of Laplace, 378, p. 318.

Scholium. The demonstration of this theorem implies that water is incompressible, and that the pressure of each particle placed on the surface is instantaneously communicated through the whole depth of the fluid to the bottom. These suppositions are not indeed strictly accurate in any case, but they introduce no sensible error when the surface of the wave similarly affected is large in comparison with the depth of the fluid. A modern author of celebrity seems to have taken it for granted that the pressure is propagated with the same velocity downwards and laterally; at least, if such is not his meaning, he has been somewhat unfortunate in the choice of his expressions; but there seems no reason whatever why water should communicate force more slowly when it is perfectly confined, than ice would do; and the divergence of the pressure of a certain portion of the surface of water, elevated a little, for example, above the rest, may be compared to the divergence of a sound entering into a detached chamber by an aperture of the same size with the given surface, which is probably small in comparison with its direct motion, but equally rapid, and in both cases depending on the modulus of the elasticity of the medium.

Theorem E. [L.] A wave of a symmetrical form, with a depression equal and similar to its elevation, striking against a solid vertical obstacle, will be reflected, so as to cause a part of the surface, at the distance of one fourth of its whole breadth, to remain at rest; and if there be another opposite obstacle at twice that distance, there may be a perpetual vibration between the surfaces, the middle point having no vertical motion. Dr Young's Natural Philosophy, vol. i. p. 289, 777.

Scholium 1. The elevation and depression of a spheroid, compared with the surface of the sphere of equal magnitude, exhibits a symmetrical wave in the sense of the proposition; and it is not necessary that the shores should be very rocky or perpendicular, in order to produce a strong reflection; for even the vibration of the water in the bottom of a common hemispherical basin is considerably permanent.

Corollary 3. The vibrations of the water supposed to be contained in a canal, following the direction of the equator, and 90° in length, would be synchronous with the passage of a wave 180° in breadth, over any point of a canal of the same depth, and surrounding the whole globe.

Scholium 2. It has been usual to consider the elevation of the tides as identical with that of an oblong spheroid, measured at its vertex, and therefore as amounting to twice as much as the depression of the same spheroid at the equator, considered in relation to the mean height belonging to a sphere of the same magnitude; but the supposition is by no means applicable to the case of a globe covered partially and irregularly with water, so that in almost all cases of actual tides, the elevation must be considered as little if at all greater than the depression, as far as this cause only is concerned; there are, however, some other reasons to expect that the elevation of the great wave might often arrive at a distant port in somewhat greater force than the depression.

Theorem F. [K.] The oscillations of the sea and of lakes, constituting the tides, are subject to laws exactly similar to those of pendulums capable of performing vibrations in the same time, and suspended from points which are subjected to compound regular vibrations, of which the constituent periods are completed in half a lunar and half a solar day [or in some particular cases a whole day].

Supposing the surface of the sea to remain at rest, each point of it would become alternately elevated and depressed, in comparison with the situation in which it might remain in equilibrium; its distance from this situation varying according to the regular law of the pendulum (see Theorem B); and, like all minute vibrations, it will be actuated by forces indirectly dependent on, and proportional to, this distance; so that it may be compared to a pendulous body remaining at rest in the vertical line, about which its point of suspension vibrates, and will consequently follow the motion of the temporary horizon, in the same manner as the pendulum follows the vibration of its point of suspension, either with a direct or a retrograde motion, according to circumstances, which will be hereafter explained; the operation of the forces concerned being perfectly analogous, whether we consider the simple hydrostatic pressure depending on the elevation, or the horizontal pressure derived from the inclination of the surface, or the differential force immediately producing elevation and depression, depending on the variation of the horizontal pressure, and proportional to the curvature of the surface. It becomes therefore necessary, for the theory of the tides, to investigate minutely the laws of these compound and compulsory vibrations, which, together with the resistances affecting them, will be the subject of the next section. Sect. III.—Of the Effects of Resistance in Vibrating Motions, whether Simple or Compound.

Theorem G. If \( dw + Ads + Bds + Duds = 0 \), we have \( e^{Dx} \left( w + \frac{B}{D} s + \frac{AD - B}{DD} \right) = c \); \( hle \) being \( = 1 \).

Scholium. For the better understanding of the mode of investigation which will be employed in these propositions, it will be proper to premise some remarks on the investigation of fluxional equations, by means of multipliers. A person unacquainted with the language of modern mathematicians, would naturally understand by a "criterion of integrability," some mode of distinguishing an expression that would be integrated, from one that was untractable; while, in fact, this celebrated criterion relates only to the accidental form in which the expression occurs, and not to its essential nature. If we take, for instance, the well-known case of the fluxion of \( \frac{x}{y} = \ln x - \ln y \), we have

\[ \frac{dx}{x} - \frac{dy}{y} = \frac{ydx - xdy}{xy}, \]

and making this \( = 0 \), we have also \( ydx - xdy = 0 \); and this expression no longer fulfils the conditions of integrability, until we multiply it again by \( \frac{1}{xy} \) and restore it to its perfect form. The direct investigation of such a multiplier is generally attended by insuperable difficulties; and the best expedient, in practical cases, is to examine the results of the employment of such multipliers as are most likely to be concerned in the problem, with indeterminate co-efficients, and to compare them with the equations proposed. In common cases, the finding of fluents, when only one variable quantity is concerned, requires little more than the employment of a table of fluents or integrals such as that of Meier Hirsch; and the truth of the solution is in general tested at once, for each case, by taking the fluxion of the quantity inserted in the table; but for the separation of different variable quantities, where they are involved with each other, the employment of proper multipliers is one of the most effectual expedients; and it is still more essential to the solution of equations between fluxions of different orders, or their co-efficients. Such equations require in general to be compared with some multiple of the exponential quantity \( e^{wt} \), which affords fluxions of successive orders, that have simple relations to each other, especially when \( dt \) is considered as constant. The multiples of sin, \( Ck \), and cos, \( Ct \), are also very useful in such investigations, and for a similar reason; but the solutions that they afford are commonly less comprehensive than the former, though they are often simpler, and more easily obtained. It is not however necessary that the exponent of the multiplier should flow uniformly, as will appear from the first example of a problem which has been solved by Euler in his Mechanics: the subsequent examples will possess somewhat more of novelty.

Demonstration. The fluxion of \( e^{wt} (w + ps + q) \) is

\[ e^{wt} (dw + pds + (uw + nps + nq) ds) = e^{wt} dw + (p + nq) \]

\( ds + ndsds + ndeds \); and comparing with this \( e^{wt} (dw + Ads + Bds + Duds) \), we have \( n = D \), \( np = B \), and \( p = \frac{B}{n} = \frac{B}{D} \); and, lastly, \( p + nq = A \), \( q = \frac{A - p}{n} = \frac{AD - B}{DD} \); consequently the fluxion of \( e^{wt} \left( w + \frac{B}{D} s + \frac{AD - B}{DD} \right) \) is equal to nothing, and that quantity is constant, or equal to \( c \).

Example. Let the given equation be that of a cycloidal pendulum, moving with a resistance proportional to the square of the velocity, or \( \frac{dds}{dt} + Bs - D \frac{ds^2}{dt^2} = 0 \).

Scholium 2. The space \( s \) being supposed to begin at the lowest point of the curve, the fluxion \( ds \) is negative during the descent on the positive side, and the force \( dds \) is consequently negative, and equal, when there is no resistance, to \( Bs \), \( B \) being a positive co-efficient, equivalent, in the case of gravitation, to \( \frac{2g}{l} \) or \( \frac{32}{l} \), \( l \) being the length of the pendulum, and \( g \) the descent of a falling body in the first second. The co-efficient \( -D \) is negative, because the resistance acts in a contrary direction to that of the force \( Bs \), as long as \( s \) remains positive, and coincides with it on the negative side. But in the return of the pendulum the signs are changed, so that the equation can only be applied to a single vibration; since the two forces in question oppose each other in the same points of the curve in which they before agreed, while the square \( \frac{ds^2}{dt^2} \) must always remain positive.

Solution. If we multiply the given equation by \( ds \), and make the square of the velocity, or \( vv = w = \frac{ds^2}{dt^2} \) we have \( ds \frac{dds}{dt} + Bds - D \frac{ds^2}{dt^2} ds = 0 = \frac{1}{2} dw + Bds - Duds \), and \( dw + 2Bds - 2Duds = 0 \); which, compared with the theorem, gives us \( 0 \) for \( A \), \( 2B \) for \( B \), and \( -2D \) for \( D \); and the solution becomes

\[ e^{-2Dt} \left( w - \frac{B}{D} s - \frac{B}{2DD} \right) = c, \]

or \( w = \frac{B}{D} s + \frac{B}{2DD} + ce^{2Ds} \); and if \( w = 0 \) when \( s = \lambda \), we have \( \frac{B}{D} \lambda + \frac{B}{2DD} + ce^{2Ds} = 0 \), or, putting \( \frac{B}{D} \lambda + \frac{B}{2DD} = \beta \), \( \beta + ce^{2Ds} = 0 \),

and \( c = -\beta e^{-2Ds} \); \( \beta \) being also \( = \frac{B}{2DD} \), if \( \gamma = 1 + 2D \lambda \). We may also substitute \( \epsilon \) for \( \lambda - s \), and \( ce^{2Ds} = -\beta e^{2D(s-\lambda)} \), will become \( -\beta e^{-2Ds} \), and \( w = \frac{B}{D} s + \frac{B}{2DD} - \beta e^{-2Ds} = \frac{B}{2DD} (1 - \gamma e^{-2Ds} + 2Ds) \).

Now \( e^{-2Ds} = 1 - 2Ds + 2D^2s^2 - \frac{4}{3}D^3s^3 + \frac{8}{5}D^4s^4 - \ldots \);

and \( (1 + 2D \lambda) e^{-2Ds} = 1 + 2D \lambda - 2Ds - 4D^2s + 2D^3s^2 - 4D^4s^3 - \ldots \); whence \( w = \frac{B}{2DD} \)

\[ (2Ds - 2D(\lambda - s) + 4D^2s - 2D^3s^2 + \ldots) = \frac{B}{2DD} \]

\[ (4D^2s^2 - 2D^3s^3 - 4D^4s^4 \ldots) = B(2s - \gamma^2 + \frac{2}{3}D^2s^2 - \frac{1}{3}D^3s^3 + \ldots). \]

Corollary 1. From this solution we obtain the point at which the velocity is greatest; and, by reversing the equation, we may also find the extent of the vibration. For when \( dw = 0 \), we have \( Bds = Duds \), and \( Bs = Ds \), which is the obvious expression of the equality of the re- Tides.

Putting the greatest value of \( w = x \), and the corresponding value of \( s = z \), we have \( z = \frac{B}{D} + \frac{B}{2DD} + ce^{2D} = \frac{B}{D} \); since \( B = D \), and \( \frac{B}{2DD} = -ce^{2D} = \frac{B}{2DD} e^{2D(1-\lambda)} \); whence \( \frac{1}{\gamma} = e^{2D(\lambda-\epsilon)} \), and \( \gamma = e^{2D(\lambda-\epsilon)} \); consequently \( hly = 2D(\lambda-\epsilon) = hl(1+2D\lambda) \), and \( \frac{1}{2D} \left( 2D^2 - \frac{8}{3} D^3 + \cdots \right) = D^2 - \frac{4}{3} D^3 + \cdots \).

And since \( z = \frac{B}{D} \), we have \( z = \frac{B}{2DD} (2D\lambda - hl [1 + 2D\lambda]) \).

Lemma. For the reversion of a series, or of a finite equation, if \( z = ax + bx^2 + cx^3 + \cdots \), we have \( z = \frac{1}{a} z - \frac{b}{a^2} z^2 + \frac{2b^2 - ac}{a^3} z^3 - \frac{5b^3 - 5abe + a^2d}{a^4} z^4 + \cdots \).

The proof of this well-known formula is the most readily obtained by means of a series with indeterminate co-efficients, such as \( z = Az + Bz^2 + \cdots \), which, by actual involution, and by comparison with the proposed series, will give the required values of the co-efficients, as expressed in this Lemma.

Corollary 2. When \( w = 0 \), we obtain from its value, divided by \( B \), the equation \( 2\lambda = \gamma - \frac{2}{3} D^2 \gamma + \frac{1}{3} D^3 \gamma^2 - \cdots \); and, by reversing this series, we have \( \sigma = \frac{2\lambda}{\gamma} + \frac{8}{3} D^2 \gamma + \frac{40}{9} D^3 \gamma^3 \cdots \), or \( \sigma = 2\lambda - \frac{4}{3} D^2 \gamma + \cdots \); the difference of the arcs of descent and ascent being \( \frac{4}{3} D^2 \gamma \), and the difference of two successive vibrations \( \frac{8}{3} D^2 \gamma \), when the resistance is very small; this difference being also \( \frac{8}{3} \gamma \); so that the displacement of the point of greatest velocity is \( \frac{3}{8} \) of the difference of the successive vibrations.

Scholium 2. If \( K \) be the value of \( w \) when \( Dw \) would be equal to the force of gravity, and \( DK = Bl = 2g \), we have \( D = \frac{2g}{K} \), or \( H \) being the height from which a body must fall to acquire the velocity \( \sqrt{K} \), since \( K = 4gH \), \( D = \frac{1}{2H} \), and \( 2D = \frac{1}{H} \).

Scholium 3. It is natural to imagine that we might obtain the time from the equation expressing the velocity in terms of the space, if we merely expanded the value of \( \frac{1}{\sqrt{w}} \) into a new series, by means of the Newtonian theorem; but the fluents thus obtained for the expression of the time are deficient in convergency; and a similar difficulty would occur if we expressed \( s \) in terms of \( w \) by reversing the series, and divided its fluxion by \( \sqrt{w} \). The ingenuity of Euler has, however, devised a method of avoiding these inconveniences, by supposing the time to begin at the point where the velocity is a maximum; and it will be necessary in this investigation, to follow his steps, with some slight variations.

Corollary 3. In order to find the time of vibration, we take \( s = r \), and \( x = w = z \), then \( s = r + \epsilon, \epsilon = \lambda - \epsilon - r, w = 2DD(1 - \gamma e^{-2D(\lambda - \epsilon - r)} + 2D[\epsilon + r]) \), and \( z \) being \( \frac{B}{D} = -\frac{B}{2DD} - \frac{B}{D} + \frac{B}{2DD} e^{2D(\lambda - \epsilon)} e^{2Dr} \); but we have seen that \( e^{2D(\lambda - \epsilon)} = \frac{1}{\gamma} \), and \( z \) becomes \( \frac{B}{2DD} + \frac{B}{D} - \frac{B}{2DD} e^{2Dr} = \frac{B}{D} - \frac{B}{2DD} (2Dr + 2D^2r^2 + \frac{4}{3} D^3r^3 + \frac{8}{12} D^4r^4 + \cdots) = \frac{B}{D} - \frac{B}{D} - Br^2 - \frac{2}{3} BD^2r^3 - \cdots \), and \( \frac{z}{B} = r^2 + \frac{2}{3} Dr^3 + \frac{1}{3} D^2r^4 + \cdots \).

In order to reverse this series, we must put \( \frac{z}{B} = y^2 \), and \( r = ay + by^2 + cy^3 + \cdots \); and by substituting the powers of this series for those of \( r \) in the value of \( \frac{z}{B} = y^2 + ry^3 + ry^4 + \cdots \), we find \( a = 1, b = -\frac{1}{2} r, c = \frac{5}{8} r^2 - \frac{1}{6} r^3 \cdots \); and \( r = y - \frac{1}{3} Dy^2 + \frac{1}{6} Dy^3 - \cdots \). Hence \( dr = dy - \frac{2}{3} Dydy + \frac{1}{3} Dy^2dy - \cdots \); and this fluxion, divided by the velocity \( e = \sqrt{(x-z)} \), will be the fluxion of the time; or, since

\[ d\frac{z}{B} = 2ydy \quad \text{and} \quad dy = \frac{dz}{2By} = \frac{dz}{2\sqrt{B}\sqrt{x-z}} \]

\[ dt = \frac{1}{B} \cdot \frac{dz}{2\sqrt{(x-z)}} - \frac{D}{SB} \sqrt{(x-z)} + \frac{D^2}{6B\sqrt{B}} \]

\[ \frac{dz}{\sqrt{(x-z)}} - \cdots \]; and the fluent becomes \( t = \frac{1}{2\sqrt{B}} \arctan \frac{2z}{x} - \frac{2D}{2B} \sqrt{(x-z)} + \frac{D^2}{6B\sqrt{B}} \left( \frac{1}{2} \arctan \frac{2z}{x} - \sqrt{(x-z)} \right) \cdots \); the value of which, taken from \( z = 0 \) to \( z = z_1 \), is \( 2 \frac{\pi}{\sqrt{B}} - \frac{2D}{3B} \sqrt{x} + \frac{D^2}{3B\sqrt{B}} \left( \frac{1}{2} \pi \right) \cdots \). If we now make \( r \) negative, for the ascent of the pendulum, the co-efficients \( r, n, \ldots, b, d, \ldots \), will change their signs, and the value of \( t \) will be \( \frac{\pi}{2\sqrt{B}} + \frac{2D}{3B} \sqrt{x} + \frac{D^2}{12B\sqrt{B}} \pi + \cdots \), the sum of both being \( \frac{\pi}{\sqrt{B}} + \frac{D^2\pi}{6B\sqrt{B}} + \cdots \), which is the time of a complete vibration, and the difference \( \frac{4D}{3B} \sqrt{x} + \cdots \).

The effect of the resistance on the whole time involves, therefore, only the second and the higher powers of the coefficient of the resistance \( D \); and it also disappears with the arc, as \( x \), the square of the greatest velocity, becomes inconsiderable with respect to the velocity itself, and to the time \( \frac{\tau}{\sqrt{B}} \).

Theorem H. If \( \frac{d^2s}{dt^2} + A \frac{ds}{dt} + Bs = 0 \), \( dt \) being con- must fall to gain the velocity \( k \), since \( k = \frac{k^2}{4g} A^2 = \frac{Ag}{kk} = \frac{g}{k} \).

Hence it follows, that when \( A^2 = 4B \), which is the time of the possibility of alternate vibrations, \( g = \frac{8g}{T} \), and \( h = \frac{1}{8} \); the resistance becoming equal to the weight when the body has fallen freely through one eighth of the length of the pendulum.

Case ii. Supposing now the resistance to be more moderate, and \( \frac{1}{4}A^2 \) to be less than \( B \), and making \( B - \frac{1}{4}A^2 = C^2 \); we shall have \( \sqrt{\left(\frac{1}{4}A^2 - B\right)} = \sqrt{\left(-C^2\right)} = \sqrt{-1}C \); the solution of the equation, \( \frac{d^2s}{dt^2} + A \frac{ds}{dt} + Bs = 0 \), will then be

\[ d\left(e^{\frac{1}{4}At} \pm \sqrt{-1}Ct \left[ \frac{ds}{dt} + \frac{1}{2}As \mp \sqrt{-1}Cs \right] \right) = 0; \]

whence, by taking the two different values in succession, and adding together their halves, we obtain

\[ d\left(e^{\frac{1}{4}At} \left[ e^{\sqrt{-1}Ct} \pm e^{-\sqrt{-1}Ct} \right] \right) = 0; \]

or,

\[ \left( ds + \frac{1}{2}As \right) + \frac{e^{\sqrt{-1}Ct} - e^{-\sqrt{-1}Ct}}{2\sqrt{-1}} Cs = ce^{-\frac{1}{4}At}. \]

Now the imaginary exponential quantities, thus combined, are the well-known expressions for the sine and cosine of the arc \( Ct \) (Elem. Illustr. § 358); and the last equation may be written thus, \( \cos. Ct \left( \frac{ds}{dt} + \frac{1}{2}As \right) + \sin. Ct Cs = ce^{-\frac{1}{4}At}; \)

whence

\[ v = -\frac{ds}{dt} = \frac{1}{2}As + \sin. Ct Cs = \frac{ce^{-\frac{1}{4}At}}{\cos. Ct}. \]

This fluent, if \( t \) were made to begin when \( v = 0 \), would only afford us such expressions as have hitherto been found intractable; but nothing obliges us to limit the problem to this condition, and it is equally allowable to make the time \( t \) begin when \( v = \frac{1}{2}As \), the corresponding value of \( s \) being called \( s_0 \); then \( \frac{1}{2}As = v = \frac{1}{2}Ae^{-\frac{1}{4}At} \); consequently \( c = 0 \). The equation will then become \( \frac{ds}{s} + C \frac{\sin. Ct}{\cos. Ct} + \frac{1}{2}Adt = 0; \) whence

\[ \frac{ds}{s} = \frac{1}{2}As + \sin. Ct Cs = \frac{ce^{-\frac{1}{4}At}}{\cos. Ct}. \]

And, when \( t = 0 \), \( s = e^{-\frac{1}{4}At} \); or \( s = \cos. Ct \cdot e^{-\frac{1}{4}At} \); and, when \( t = 0 \), \( s = e^{-\frac{1}{4}At} \); consequently \( \frac{s}{s} = \cos. Ct \cdot e^{-\frac{1}{4}At} = \cos. Ct \left( 1 - \frac{1}{8}A^2t^2 - \frac{1}{48}A^4t^4 + \cdots \right) \). But since \( v = -\frac{ds}{dt} = \frac{1}{2}As + \sin. Ct Cs = \frac{ce^{-\frac{1}{4}At}}{\cos. Ct} \),

it follows that \( v \) must vanish whenever \( C \frac{\sin. Ct}{\cos. Ct} + \frac{1}{2}A = 0 \), or when \( \tan. Ct = \frac{A}{2C} \), that is, in the first instance, very nearly when \( Ct = \frac{A}{2C} \), and \( t = \frac{A}{2CC} \), and \( \frac{1}{2}At = \frac{AA}{6CC} \) and \( e^{-\frac{1}{4}At} = 1 + \frac{AA}{4CC} \), very nearly; so that, calling the primitive extent of the arc of vibration \( s = \lambda \), we have \( \frac{\lambda}{s} = \cos. Ct \left( 1 + \frac{AA}{4CC} \right) \); \( \cos. Ct \) Tides

being also, in this case, \( \sqrt{1 - \frac{AA}{4CC}} = 1 - \frac{AA}{8CC} \), and

\( \lambda = 1 + \frac{AA}{8CC} \), and \( \frac{5}{\lambda} = \frac{8CC}{8CC + AA} = \frac{8B - 2AA}{8B - AA} = 1 - \frac{AA}{8B - AA} \), corresponding to the verse sine of the time \( \frac{A}{2CC} \)

or to the arc \( \frac{A}{2C} \), in the circle represented by \( Ct \).

Corollary 2. It follows that both \( v \) and \( s \) must vanish continually at equal successive intervals, whenever \( ta. Ct = \frac{A}{2C} \), and when \( \cos. Ct = 0 \), respectively; the descent to the lowest point will therefore occupy the time corresponding to \( \frac{\pi}{4} + \frac{A}{2C} \), and the subsequent ascent to \( \frac{\pi}{4} - \frac{A}{2C} \); the extent of the vibrations being always proportional to \( e^{-\frac{1}{2}At} \).

Corollary 3. The greatest velocity must take place at the point where \( A \frac{ds}{dt} + Bs = 0 \), and \( AC ta. Ct + \frac{1}{2}A^2 = B \), or \( ta. Ct = \frac{B - \frac{1}{2}AA}{AC} \), and \( \cot. Ct = \frac{AC}{B - \frac{1}{2}AA} \); or, if we neglect \( A^2 \), \( \cot. Ct = \frac{A}{\sqrt{B}} = \cos. Ct = \frac{s}{c} \) very nearly.

Corollary 4. The diminution of the successive vibrations is expressed by the multiplier \( e^{-\frac{1}{2}At} \), which, when \( Ct = 2\pi \), the whole circumference, is \( 1 - \frac{A}{C} \pi \), and \( \frac{A}{C} \pi \lambda \), or \( \frac{A\pi}{\sqrt{B}} \), is the diminution of the value of \( s \) when the pendulum returns to the place from which it first set out, that is, the difference between the lengths of two vibrations, each corresponding to a semicircumference, and this difference is to \( \frac{A}{\sqrt{B}} \), or \( \frac{A}{\sqrt{B}} \lambda \), the displacement of the point of greatest velocity, which measures the greatest resistance, as \( \sigma \) to 1, or as 3:1416 to 1. We have seen that, for a resistance varying as the square of the velocity, this proportion was as 8 to 3, or as 2:667 to 1.

Corollary 5. If the pendulum be supposed to vibrate in a second, the unity of time, the diminution of the arc \( 2\lambda \) in each vibration will be \( \frac{1}{2}A \times 2\lambda \), and the successive lengths will vary as \( e^{-\frac{1}{2}A 2\lambda}, e^{-\frac{1}{2}A 2\lambda}, \) and so forth; and after the number \( N \) of vibrations, the extent of the arc will be reduced from \( 2\lambda \) to \( e^{-\frac{1}{2}NA 2\lambda} \); so that if we make \( e^{-\frac{1}{2}NA} = M \), we have \( hlM = -\frac{1}{2}NA \), and \( A = \frac{2}{N} hl \frac{1}{M} \). Thus, if in an hour the vibrations were reduced to \( \frac{2}{3} \) of their extent, which is rather more than appears to have happened in any of Captain Kater's experiments, we should have \( N = 3600 \), and \( M = \frac{2}{3} \), whence \( A = \frac{1}{1800} \times 4054651 = 0.0022526 \), and \( A^2 = 0.0000005075 \); and since \( B = \frac{32}{\pi} = 9.81 \), \( C = \sqrt{(B - \frac{1}{2}A^2)} = \sqrt{B} \sqrt{1 - \frac{AA}{4B}} = \sqrt{B} \left(1 - \frac{AA}{78.5}\right) \); the fraction being only \( 0.000000065 \); or about one second in 1600 millions, that is, in about fifty years.

Scholium 3. Although the isochronism of a pendulum, with a resistance proportional to the velocity, was demonstrated by Newton, yet Euler appears to have failed in his attempts to carry the theory of such vibrations to perfection; for he observes (Mechan. ii. p. 312), *Etsi ex his apparente, tempora tam ascensuum quam descensuum inter se esse aquatia, tamen determinari non potest, quantum sit tempus sine descensuum sine ascensuum: neque enim tempora descensuum et ascensuum inter se possint comparari.* *Equatio enim rationem inter s et u definiens ita est complicata, ut ex ea elementum temporis \( \frac{ds}{u} \), per unicam variabilen non posse exprimi.*

Scholium 4. In confirmation of the solution that has been here proposed, it may not be superfluous to show the truth of the result in a different manner. Taking \( s = e^{mt} \cos. Ct \), we have \( \frac{ds}{dt} = ce^{mt} (m \cos. Ct - C \sin. Ct) \),

and \( \frac{dds}{dt^2} = ce^{mt} (m^2 \cos. Ct - Cm \sin. Ct - Cm \sin. Ct - C^2 \cos. Ct) \); whence \( \frac{dds}{dt^2} + A \frac{ds}{dt} + Bs = ce^{mt} (m^2 \cos. Ct - 2Cm \sin. Ct - C^2 \cos. Ct + Am \cos. Ct - AC \sin. Ct + B \cos. Ct) = 0 \), and \( (m^2 - C^2 + Am + AC) \cos. Ct - (2Cm + AC) \sin. Ct = 0 \): an equation which is obviously true when the co-efficients of both its terms vanish, and \( 2Cm = -AC \), or \( m = \frac{1}{2}A \); and again \( C^2 = m^2 + Am + B = \frac{1}{2}A^2 - \frac{1}{2}A^2 + B = B - \frac{1}{2}A^2 \). The former mode of investigation is more general, and more strictly analytical; but this latter is of readier application in more complicated cases, and it will hereafter be further pursued.

Lemma. If a moveable body be actuated continually by a force equal to that which acts on a given pendulum, the body being in a state of rest when the pendulum is at the middle of its vibration, the space described in the time of a vibration will be to the length of the pendulum as the circumference of a circle is to its diameter. For the force being represented by \( \cos. Ct \) or \( \cos. x \), for the pendulum, it will become \( \sin. x \) with regard to the beginning of the supposed motion, and the velocity, instead of \( \sin. x \), becomes \( -\cos. x \), or \( 1 - \cos. x \); so that the space, instead of \( 1 - \cos. x \), is \( x - \sin. x \), which, at the end of the semivibration, is \( x = \frac{\pi}{2} \), instead of \( 1 - \cos. x = 1 \), the space described by the simple pendulum, which is equal to its length.

Scholium 5. There is a paradox in the relations of the diminution of the vibration to the distance measuring the greatest resistance, which it will be worth while to consider, in order to guard ourselves against the too hasty adoption of some methods of approximation which appear at first sight unexceptionable. The pendulum, if it set out from a state of rest at the point of greatest resistance, would perform a vibration to the extent of double the distance of that point, or \( 2 \frac{A}{\sqrt{B}} \lambda \), the initial force being measured by that distance. Now, when the resistance is very small, its magnitude may be obtained without sensible error from the velocity of the pendulum vibrating without resistance at the corresponding part of the arc; and the velocity may be supposed to vary as \( \sin. Ct \), and the resistance, in the case of this proposition, as \( \sin. Ct \) or \( \sin. x \) also. Hence it may be inferred by means of the Lemma, that the whole diminution of the space will be to \( \frac{A}{\sqrt{B}} \lambda \) as \( \sigma \) to 1, or that it will be equal to \( \frac{A}{\sqrt{B}} \pi \lambda \), which has already been found to be the actual difference of two successive semivibrations. The accuracy of this result, however, must depend on the mutual compensation of its errors; for the approximation supposes, that if the resistance vanished at the lowest point, the subsequent retardation would be such as to diminish the space by the effect of the diminution of the velocity acting uniformly through the remainder of the vibration, while in fact the diminution of the space from this cause would be simply equal to a part of the arc proportional to the diminution of the velocity, since the arc of ascent is simply as the velocity at the lowest point. Hence it is obvious, that the effects of the resistance are too much complicated with the progress of the vibration to allow us to calculate them separately; and accordingly, when the resistance is as the square of the velocity, or as $\sin^2 x$, the diminution of velocity is expressed by $\frac{1}{2} \cos x - \frac{1}{2} \sin x \cos x$, and that of the space by $\frac{1}{2} \sin x - \frac{1}{2} \sin^2 x$, which, at the end of a vibration, becomes $\frac{1}{2} \sin x$ instead of $\frac{1}{2} \sin x$, that is, since the distance of the point of greatest velocity is here $s = D\lambda^2$, $\frac{1}{2} D\lambda^2 = 2.467 D\lambda^2$, while the more accurate mode of computation has shown that the true diminution of the space is $2.667 D\lambda^2$. (Theorem G.) If we choose to pursue the mode of approximation here suggested, with accuracy, it would be necessary to consider the resistance as a periodical force acting on a pendulum capable of synchronous vibration, as hereafter in Theorem K, Schol. 1.

**Theorem J.** If $\frac{d^2 s}{dt^2} + Bs + M \sin Ft = 0$, we may satisfy the equation by taking $s = \sin (\sqrt{B}t) + \frac{M}{FF-B} \sin Ft$.

**Demonstration.** The value of $s$ here assigned gives us $\frac{ds}{dt} = \sqrt{B} \cos \sqrt{B}t + \frac{MF}{FF-B} \cos Ft$, and $\frac{d^2 s}{dt^2} = -B \sin \sqrt{B}t - \frac{MFF}{FF-B} \sin Ft$; so that $\frac{d^2 s}{dt^2} + Bs = B \sin \sqrt{B}t + \frac{MB}{FF-B} \sin Ft - B \sin \sqrt{B}t - \frac{MFF}{FF-B} \sin Ft = \frac{MB-MFF}{FF-B} \sin Ft = -M \sin Ft$.

**Corollary 1.** If, in order to generalize this solution, we make $s = a \sin \sqrt{B}t + b \cos \sqrt{B}t + c \sin Ft + d \cos Ft$, we may take any quantities at pleasure for $a$ and $b$, according to the conditions of the particular case to be investigated; but $c$ must be $= 0$; that is, the motion will always be compounded of two vibrations, the one dependent on the length of the pendulum, or on the time required for the free vibration, indicated by $\sqrt{B}t$, the other synchronous with $Ft$, the period of the force denoted by $M$; the latter only being limited to the condition of beginning and ending with the periodical force.

**Corollary 2.** In the same manner, it may be shown that the addition of any number of separate periodical forces, indicated by the terms $MP \sin Ft$, $MP' \sin Ft'$, etc., will add to the solution the quantities $\frac{MP}{FF-B} \sin Ft$, $\frac{MP'}{FF'-B} \sin Ft'$, and so forth.

**Example 1.** Supposing a pendulum to be suspended on a vibrating centre, and to pass the vertical line at the same moment with the centre, we may make $a$ and $b = 0$, and $s = \frac{M}{FF-B} \sin Ft$ only; the vibration being either direct or reversed, according as $F$ is less or greater than $\sqrt{B}$, or than $\sqrt{\frac{32}{l}}$, which determines the spontaneous vibration of the pendulum.

**Example 2.** But if the ball of the pendulum be supposed to begin its motion at the moment that the centre of suspension passes the vertical line, we must make $s = \frac{M}{FF-B} \sin Ft - \cos \sqrt{B}t$; and the subsequent motion of the pendulum will then be represented by the sum of the sines of two unequal arcs in the same circle; and if these arcs are commensurate with each other, the vibration will ultimately acquire a double extent, and nearly disappear in a continued succession of periods, provided that no resistance interfere. And the consequences of any other initial conditions may be investigated in a manner nearly similar. Thus, if the time of free vibration, under these circumstances, were $\frac{1}{2}$ of the periodical time, the free vibration, in which the motion must be supposed initially retrograde, in order to represent a state of rest by its combination with the fixed vibration, would have arrived at its greatest excursion forwards, after three semivibrations, at the same moment with the fixed vibration, and after three complete vibrations more would be at its greatest distance in the opposite direction, so as to increase every subsequent vibration equally on each side, and permanently to combine the whole extent of the separate arcs of vibration. But in this and in every other similar vibration, beginning from a state of rest in the vertical line, that is, at the point where the periodical force is evanescent, the effect of the free or subordinate vibration with respect to the place of the body will obviously disappear whenever an entire number of semivibrations has been performed.

**Corollary 3.** The paradox stated in the fourth scholium on the last theorem may be illustrated by means of this proposition, and will serve in its turn to justify the mode of computation here employed in a remarkable manner. It has been observed in Nicholson's Journal for July 1813, that the mode of investigating the effects of variable forces, by resolving them into parts represented by the sines of multiple arcs, and considering the vibrations derived from each term as independent in their progress, but united in their effects, may be applied to the problem of a pendulum vibrating with a resistance proportional to the square of the velocity; and that for this purpose the square of the sine may be represented by the series $\sin^2 x = 8484 \sin x - 1696 \sin 3x + 2444 \sin 5x - 60813 \sin 7x + 9029 \sin 9x - 9013 \sin 11x - \ldots$. Now, if we employ this series for resolving the resistance supposed in Theorem G into a number of independent forces, the greatest resistance being measured by $\frac{A}{\sqrt{B}}$, we shall have $8484 \frac{A}{\sqrt{B}}$ for the part supposed to be simply proportional to the velocity, whence, from Theorem H, we have $8484 \frac{A}{\sqrt{B}}$ for the corresponding diminution of the vibration; that is, $2.6653 \frac{A}{\sqrt{B}}$. But it has been observed, in the preceding corollary, that the place of the pendulum will not be at all affected by any subordinate vibration after any entire number of complete semivibrations; and the slight effect of the velocity left in consequence of these subordinate vibrations may here be safely neglected, so that $2.6653 \frac{A}{\sqrt{B}}$ may be considered as the whole effect of the resistance with respect to the space described, which differs only by $\frac{1}{2000}$ of the whole from $2666 \frac{A}{\sqrt{B}}$, the result of the more direct computation of Theorem G.

**Scholium.** An experimental illustration of the accuracy of the theorem may be found in the sympathetic vibrations of clocks, and in that of the inverted pendulum invented by Mr Hardy, as a test of the steadiness of a support (art. Pendulum, vol. xvii. p. 218); for since the extent of the regular periodical vibration is measured by $\frac{M}{FF-B}$, it is evident, that however small the quantity $M$ may be, it will become very considerable when divided by $FF-B$, as $F$ and $\sqrt{B}$ approach each other; and accordingly it is observed, that when the inverted pendulum is well adjusted to the rate of a clock, there is no pillar so steady as not to communicate to it a very perceptible motion by its regular, though extremely minute, and otherwise imperceptible change of place.

**Theorem K.** In order to determine the effect of a periodical force, with a resistance proportional to the velocity, the equation $\frac{d^2s}{dt^2} + A \frac{ds}{dt} + Bs = M \sin. Gt = 0$, may be satisfied by taking $s = \alpha \sin. Gt + \beta \cos. Gt$, $\alpha$ being

$$\frac{GG - B}{(GG - B)^2 + AAGG} M,$$

and $\beta =$

$$\frac{AGM}{(GG - B)^2 + AAGG}, s$ being also $\sqrt{(\alpha^2 + \beta^2)} \sin.$

$$\left(Gt + \arctan \frac{\beta}{\alpha}\right) = \sqrt{[(GG - B)^2 + AAGG]} \sin.$$

$$\left(Gt - \arctan \frac{AG}{B - GG}\right).$$

Since $s = \alpha \sin. Gt + \beta \cos. Gt$, $\frac{ds}{dt} = \alpha G \cos. Gt - \beta G \sin. Gt$, and $\frac{d^2s}{dt^2} = -\alpha G^2 \sin. Gt - \beta G^2 \cos. Gt = -G^2s$;

consequently the equation becomes $(B - G^2)(\alpha \sin. Gt + \beta \cos. Gt) + \alpha AG \cos. Gt - \beta AG \sin. Gt + M \sin. Gt = 0$, and $(B - G^2)\alpha - \beta AG + M = 0$, and $(B - G^2)\beta + \alpha AG = 0$; whence $\beta = \frac{AG}{GG - B}$ also $\beta = \frac{AG}{GG - B}$ and $(G^2 - B)M - (G^2 - B)^2\alpha = \alpha AG^2$; consequently $\alpha = \frac{(GG - B)M}{(GG - B)^2 + AAGG}$ and $\beta = \frac{AGM}{(GG - B)^2 + AAGG}$. And since, in general, if $b = \tan b$, $\sin x + b \cos x = \sqrt{(1 + b^2)} \sin(x + b)$; $\sin(x + b)$ being $\sin x \cos b + \sin b \cos x = \cos b \sin x + \tan b \cos x$, and therefore $\sin x + \tan b \cos x = \frac{\sin(x + b)}{\cos b} = \sin(x + b) \sec b = \sin(x + b) \sqrt{(1 + b^2)}$:

it follows that $\alpha \sin. Gt + \beta \cos. Gt = \alpha (\sin. Gt + \arctan \frac{\beta}{\alpha})$

$$\sqrt{1 + \frac{\beta^2}{\alpha^2}};$$ and $\alpha \sqrt{1 + \frac{\beta^2}{\alpha^2}} = \sqrt{(\alpha^2 + \beta^2)} =$

$$\frac{AGM}{(GG - B)^2 + AAGG} M = \sqrt{[(GG - B)^2 + AAGG]} \sin.$$

**Corollary.** If we put $M \cos. Gt$ instead of $M \sin. Gt$, we shall have $s = \alpha' \sin. Gt + \beta' \cos. Gt$; $\alpha'$ being $\beta' =$

$$\frac{B - GG}{(GG - B)^2 + AAGG} M,$$

and $s = \sqrt{(\alpha'^2 + \beta'^2)} \sin.$

$$\left(Gt + \arctan \frac{\beta'}{\alpha'}\right) = \sqrt{[(GG - B)^2 + AAGG]} \sin.$$

**Scholium 1.** Supposing $B$ to approach very near to $G^2$, a case very likely to occur in nature, because the effects which are produced, where it is found, will predominate over others, on account of the minuteness of the divisor; we may neglect the part of the denominator $(G^2 - B)^2$, in comparison with $A^2G^2$, and the co-efficient determining $s$ will then become $\frac{M}{AG}$, the extent of the vibrations being inversely as $A$ the co-efficient of the resistance; and, indeed, when the whole force of the periodical vibration is expended in overcoming a resistance proportional to the velocity, it may naturally be imagined that the velocity should be inversely as the resistance. It follows also from the proposition, that in this case the arc ta. $\frac{AG}{B - GG}$ approaching to a quadrant, the greatest excursions of the periodical motion and of the free vibration will differ nearly one fourth of the time of a complete vibration from each other.

**Scholium 2.** Since $s$ is a line, and $B$ its numerical coefficient, making it represent a force, and since $\sin. Gt$ is properly a number also, the co-efficient $M$, both here and in Theorem J, must be supposed to include another linear co-efficient, as $\mu$, which converts the sine into a line, to be added to $s$, the distance from the middle point; that is, $M$ must be considered as representing $Bu$, in which $\mu$ is the true extent of the periodical change of the centre of suspension, and $B = \frac{2g}{l}$, as in other cases; so that $M$ is

$$= \frac{2g}{l} \mu = 32 \frac{g}{l},$$

and $\mu = \frac{MI}{2g} = \frac{1}{32} MI$.

**Corollary.** In order to obtain a more general solution of the problem, we may combine the periodical motions thus determined with the free vibrations, as computed in Theorem H, the different motions, as well as the resistances, being totally independent of each other; but the most interesting cases are those which are simply periodical, the free vibration gradually diminishing with the multiplier $e^{-mt}$ and ultimately disappearing.

**Theorem L.** If there are several periodical forces, the equation $\frac{d^2s}{dt^2} + A \frac{ds}{dt} + Bs + M \sin. Gt + N \sin. Ft + ... = 0$, may be satisfied by taking $s = \alpha \sin. Gt + \beta \cos. Gt + c' \sin. Ft + \beta' \cos. Ft + ... = \sqrt{[(G^2 - B)^2 + A^2G^2]} \sin.$

$$\left(Gt - \arctan \frac{AG}{B - GG}\right) + \sqrt{[(F^2 - B)^2 + A^2F^2]} \sin.$$

$$\left(Ft - \arctan \frac{AF}{B - FF}\right) + ...$$

For, the equations expressing the space described being simply linear, the different motions and resistances are added or subtracted without any alteration of the respective relations and effects.

**Scholium.** A free vibration may also be combined with this compound periodical vibration, by means of Theorem H; but it will gradually disappear by the effect of the resistance.

**Lemma.** For the addition of the arcs $a$ and $b$, beginning with the well-known equation $\sin(a + b) = \sin a \cos b + \cos a \sin b$, we have, by addition, $\sin(a + b) + \sin(a - b) = 2 \sin a \cos b$, and $\sin a \cos b = \frac{1}{2} \sin(a + b) + \frac{1}{2} \sin(a - b)$. Then, if $c = b + 90^\circ$, $\cos b = \sin c$, whence $\sin a \sin c = \frac{1}{2} \sin(a + c - 90^\circ) + \frac{1}{2} \sin(a - c + 90^\circ)$; but $\sin(x + 90^\circ) = \cos x$, and $\sin(x - 90^\circ) = -\cos x$, consequently $\sin a \sin c = \frac{1}{2} \cos(a - c) - \frac{1}{2} \cos(a + c)$. Again, if \( c = a - 90^\circ \), \( \cos c = \sin a \), and \( \cos c \cos b = \frac{1}{2} \sin (a + b) + \frac{1}{2} \sin (a - b) = \frac{1}{2} \sin (c + 90^\circ + b) + \frac{1}{2} \sin (c + 90^\circ - b) = \frac{1}{2} \cos (c + b) + \frac{1}{2} \cos (c - b) \). Also, since \( \cos a \cos b = \frac{1}{2} \cos (a + b) + \frac{1}{2} \cos (a - b) \), and \( \sin a \sin b = \frac{1}{2} \cos (a - b) - \frac{1}{2} \cos (a + b) \), we have, by subtraction, \( \cos (a + b) = \cos a \cos b - \sin a \sin b \), and, by addition, \( \cos (a - b) = \cos a \cos b + \sin a \sin b \).

Corollary. If \( a + b = c \) and \( a - b = d \), \( \cos c + \cos d = 2 \cos \frac{c + d}{2} \cos \frac{c - d}{2} \); and \( \cos d - \cos c = 2 \sin \frac{c + d}{2} \sin \frac{c - d}{2} \); and \( \sin a - \sin b = \sin a + \sin (-b) = 2 \sin \frac{a - b}{2} \cos \frac{a + b}{2} \).

Theorem M. The equation \( \frac{d^2 s}{dt^2} + A \frac{ds}{dt} + Bs + R \sin Ft \sin Gt = 0 \), may be solved by taking \( s = a \sin [(F - G)t + p] - \beta \sin ((F + G)t + q) \); \( a \) being

\[ \sqrt{(F - G)^2 - B^2} + A(F - G) \]

\[ \beta = \sqrt{(F + G)^2 - B^2} + A(F + G) \]

\( = \arctan \frac{B - (F - G)}{A(F - G)} \), and \( q = \arctan \frac{B - (F + G)}{A(F + G)} \).

For since \( Ft \sin Gt = \frac{1}{2} \cos (F - G)t - \frac{1}{2} \cos (F + G)t \), the equation becomes \( \frac{d^2 s}{dt^2} + A \frac{ds}{dt} + Bs - R \cos (F + G)t + \frac{1}{2} R \cos (F - G)t = 0 \); whence we obtain the solution by comparison with Theorem K and its corollary.

IV.—Astronomical Determination of the Periodical Forces which Act on the Sea or on a Lake.

In order to compute, by means of the theory which has been laid down in the two preceding sections, the primitive forces of any sea or any portion of the ocean, we must compare its spontaneous oscillations with those of a narrow prismatic canal, situated in a given direction with respect to the meridian, which in general must be that of the greatest length of the sea in question, neglecting altogether the actual breadth of the sea, which, if considerable, may require to have its own distinct vibrations compounded with those of the length, each being first computed independently of the other. Now, supposing the time required for the principal spontaneous oscillation of the sea or lake to be known, we must find the length of the synchronous pendulum, and taking \( B = \frac{2g}{l} = \frac{32}{l} \), we must next find a series for expressing the force in terms of the sine, or cosines of multiple arcs, increasing uniformly with the time.

Now the force is measured, for the direction of the meridian of the spheroid of equilibrium, by \( \sin \cos z \) (Theorem A), \( z \) being either the zenith distance or the altitude; and it is obvious that, when the canal is situated obliquely with respect to the meridian of the spheroid, the inclination of the surface, and with it the force, will be diminished as the secant of the obliquity increases, or as the cosine of the obliquity diminishes; so that the force will vary as \( \sin \cos Alt \sin (Az + Decl) \); and it is obvious that this force will vanish both when the luminary is in the horizon, and when it is in the vertical circle, perpendicular to the direction of the canal; that is, if we consider the force acting horizontally on a particle at the middle of the length of the given canal; and the same force may be considered as acting vertically, with a proper reduction of its magnitude, at the end of the canal; for the horizontal oscillations at the middle must obviously follow the same laws as the vertical motions at the end.

The case, however, of a canal running east and west, admits a very remarkable simplification; and since it approaches nearly to that of an open ocean, which has been most commonly considered, it will be amply sufficient for the illustration of the present theory. For, in general, \( \sin Az = \cos Decl \sin Hor < \cos Alt \), and the expression, \( \sin \cos Alt \sin Az \), becomes in this case \( \sin Alt \cos Decl \sin Hor < \cos Alt \), but \( \sin Alt = \sin (Lat) \sin Decl + \cos (Lat) \cos Decl \cos Hor < \), and calling \( \sin (Lat) \) for the given canal \( l \), and \( \cos (Lat) \nu \), the force becomes \( l \sin \cos Decl \sin Hor < + \nu \cos^2 Decl \sin \cos Hor < \). Now, \( \sin Decl = \cos Obl Ecl \sin Lat + \sin Obl Ecl \cos Let \sin Long \); and since \( \varphi = 1 - \frac{1}{4} \sin^2 \varphi + \frac{3}{8} \sin^4 \varphi - \frac{5}{16} \sin^6 \varphi + ... \), the true value of \( \cos Decl \) might be expressed, if required, by means of this series, and its second and fourth powers would in general be sufficient for the computation.

But it will be more convenient to suppose the sun and moon to move in the ecliptic, and the ecliptic to be at the same time so little inclined to the equator, that the longitude may be substituted for the right ascension; a substitution which will cause but little alteration in the common phenomena of the tides. Then, if the sun's longitude be \( \Theta \), and the moon's \( \Phi \), the horary angles \( t \) and \( \ell \), and the sine of the obliquity of the ecliptic \( \alpha \), we shall have \( \sin Decl = \alpha \sin \Theta \), or \( \alpha \sin \Phi \); and \( \cos Decl = 1 - \sin^2 \Theta \), \( \cos Decl + \frac{3}{8} \sin^4 Decl = 1 - \frac{1}{4} \alpha^2 \sin^2 \Theta + \frac{3}{8} \alpha^4 \sin^4 \Theta \), and \( \sin \cos Decl = \alpha \sin \Theta - \frac{1}{2} \alpha^2 \sin^3 \Theta + \frac{3}{8} \alpha^4 \sin^5 \Theta \); also \( \cos^2 Decl = 1 - \alpha^2 \sin^2 \Theta \); whence the sun's force becomes \( l \sin t (\alpha \sin \Theta - \frac{1}{2} \alpha^2 \sin^3 \Theta + \frac{3}{8} \alpha^4 \sin^5 \Theta) + \nu \cos^2 \Theta \sin 2t (1 - \alpha^2 \sin^2 \Theta) = l \sin t (\alpha \sin \Theta - \frac{1}{2} \alpha^2 \sin^3 \Theta + \frac{3}{8} \alpha^4 \sin^5 \Theta) + \nu \cos^2 \Theta \sin 2t (1 - \alpha^2 \sin^2 \Theta) \).

\( \alpha^2 \) being \( \alpha = \frac{3}{8} \alpha^3 + \frac{15}{64} \alpha^5 + ... \); or about .3645;

\[ \alpha^2 = \frac{1}{8} \alpha^3 - \frac{15}{128} \alpha^5 + ... = .0078, \quad \text{and} \quad \alpha^2 = \frac{3}{128} \alpha^5 + ... = .00002, \quad \text{and} \quad \alpha^2 = .1585. \]

But \( \sin t \sin \Theta = \frac{1}{2} \cos (t - \Theta) - \frac{1}{2} \cos (t + \Theta) \), and \( \sin 2t \cos 2\Theta = \frac{1}{2} \sin 2(t + \Theta) + \frac{1}{2} \sin 2(t - \Theta) \). Hence the sun's force becomes \( S \)

\[ S = \left( \frac{1}{2} \cos (t - \Theta) - \frac{1}{2} \cos (t + \Theta) \right) + \nu \cos^2 \Theta \left[ \frac{1}{2} \cos (t - \Theta) - \frac{1}{2} \cos (t + \Theta) \right] + \nu \cos^2 \Theta \left[ \frac{1}{2} \cos (t - \Theta) - \frac{1}{2} \cos (t + \Theta) \right] + \nu \cos^2 \Theta \left[ \frac{1}{2} \cos (t - \Theta) - \frac{1}{2} \cos (t + \Theta) \right] + \nu \cos^2 \Theta \left[ \frac{1}{2} \cos (t - \Theta) - \frac{1}{2} \cos (t + \Theta) \right] \]

and that of the moon may be expressed in the same manner, by substituting \( M, \ell, \) and \( \Phi \), for \( S, t, \) and \( \Theta \). The effect of that part of the hydraulic resistance, which is proportional to the square of the velocity, must be expressed by an approximation deduced from the periodical character of the force, as depending on that of the primitive forces concerned; taking, however, the precaution to use such expressions only as will always represent this resistance in its proper character as a retarding force: for if we simply found for it an equivalent expression, denoting accurately the square of the velocity, this square, being always positive, would imply a force acting always in the same direction. Now, we have already seen (Theorem J, Cor. 3), that $\sin^2 x$ may be considered, with respect to its principal effect, as equivalent to $8484 \sin x$: and if we neglect, in the determination of the resistance, the effect of the smaller forces, and compute only that of the principal terms $\frac{1}{2} L' \sin 2t$, and $\frac{1}{2} L' \sin 2t$, we may call the velocities depending on these forces $S \cos (2t + s')$ and $M' \cos (2t' + m')$; $S'$ and $M'$ representing not exactly the proportion of the primitive forces of the sun and moon, but that of the tides depending on their combination with the conditions of the given sea or lake. The resistance will then be as the square of $S [\cos (2t + s') + \cos (2t' + m')] + (M' - S') \cos (2t' + m')$, and when least, it will be $D (M' - S')^2$; and when greatest, $D (M' + S')^2$, the difference being $4DM'S$; so that the difference may be sufficiently represented by $4DM'S [\cos (2t + s') + \cos (2t' + m')] \times 8484$, or rather $(8484)^2$, because the value of $\cos t + \cos t = 2 \cos \frac{t + t'}{2} \cos \frac{t - t'}{2}$, which is to be squared, requires the reduction from 1 to 8484 for each of its factors; and in this manner we obtain a perfect representation of the period and quality of the resistance, and a very near approximation to its magnitude.

It will, however, be still more accurate to consider the resistance thus determined as comprehended in the value of the co-efficient $A$, substituting for it, in the case of the solar tide, $A' = A + 288 DM'$, and for the moon $A'' = A + 288 DS' + 8484 D (M' - S)$; this latter part expressing that portion of the resistance $D$ which observes the period of the lunar tide, and which may therefore be considered as added to the resistance $A$ for that tide only.

Hence, collecting all the forces concerned into a single equation, the expression will become

$$\frac{d^2s}{dt^2} + A'' \frac{ds}{dt} + B_s + S \left[ \frac{1}{2} \cos (t - \Theta) - \frac{1}{2} \cos (t + \Theta) + L \alpha' \left[ \frac{1}{2} \cos (t - 3\Theta) - \frac{1}{2} \cos (t + 3\Theta) \right] + \alpha'' \left[ \frac{1}{2} \cos (t - 5\Theta) - \frac{1}{2} \cos (t + 5\Theta) \right] + \frac{L'}{2} (1 - \alpha^2) \sin 2t + \frac{L'}{4} \alpha^2 \left[ \frac{1}{2} \sin 2(t - \Theta) + \frac{1}{2} \sin 2(t + \Theta) \right] \right] + M \left[ L \alpha' \left[ \frac{1}{2} \cos (t - \Theta) - \frac{1}{2} \cos (t + \Theta) \right] + L \alpha'' \left[ \frac{1}{2} \cos (t - 3\Theta) - \frac{1}{2} \cos (t + 3\Theta) \right] + L \alpha'' \left[ \frac{1}{2} \cos (t - 5\Theta) - \frac{1}{2} \cos (t + 5\Theta) \right] + \frac{L'}{2} (1 - \alpha^2) \sin 2t + \frac{L'}{4} \alpha^2 \left[ \frac{1}{2} \sin 2(t - \Theta) + \frac{1}{2} \sin 2(t + \Theta) \right] \right] = 0;$$

and from each of these terms the value of the corresponding pair of terms in the value of $s$ may be obtained independently, by comparison with the $M \sin Gt$ or $N \cos Gt$ of Theorem K, which gives us

$$\frac{(GG - B) \sin Gt + AG \cos Gt}{(GG - B)^2 + AAGG} = M,$$

and

$$\frac{AG \sin Gt + (B - GG) \cos Gt}{(GG - B)^2 + AAGG} = N,$$

respectively.

But without entering minutely into the effects of all the terms of the equation of the forces, it may be observed in general that their results, with regard to the space described, will not differ much from the proportion of the forces, except when their periods approach nearly to that of the spontaneous oscillation, represented by $R$. Thus, since $\frac{1}{2} \cos (t - \Theta) - \frac{1}{2} \cos (t + \Theta)$ is the representative of $\sin t \sin \Theta$, and since these terms will afford results in the form $\frac{1}{2} \alpha \cos (t - \Theta) + \frac{1}{2} \beta \sin (t - \Theta)$, and of $\frac{1}{2} \alpha \cos (t + \Theta) + \frac{1}{2} \beta \sin (t + \Theta)$, and if we neglected the slight difference of $\alpha$ and $\alpha'$, which is that of $\left(1 - \frac{\Theta}{t}\right)^2 - B$, and $\left(1 + \frac{\Theta}{t}\right)^2 - B$, $\frac{\Theta}{t}$ being $\frac{1}{365.254}$ only, we should have $\frac{1}{2} \alpha [\cos (t - \Theta) - \cos (t + \Theta)] + \frac{1}{2} \beta [\sin (t - \Theta) - \sin (t + \Theta)] = \alpha \sin t \sin \Theta + \beta \cos t \sin \Theta = \sin \Theta (\alpha \sin t + \beta \cos t)$; which is the same as if we considered the effect of the force $s$ separately, and afterwards reduced it in the proportion of $\sin \Theta$. Hence it is obvious, that for all modifications of the forces greatly exceeding in their periods the period of spontaneous oscillation, the effects may be computed as if the forces were exempt from those modifications, and then supposed to be varied in the same proportion as the forces; but we cannot be quite certain of the magnitude of the error thus introduced, unless we know the exact value of $B$, which determines the time of spontaneous oscillation.

Considering, therefore, in this simple point of view, the correct expression of the force $L \sin \Theta$, cos. Decl. sin. Hor. $< + L' \cos^2 Decl. \sin \cos. Hor. <$, or $\frac{1}{2} \sin 2 Decl. \sin. Hor. < + \frac{1}{2} \cos^2 Decl. \sin. 2 Hor. <$, we may observe that the phenomena for each luminary will be arranged in two principal divisions, the most considerable being represented by $\frac{1}{2} L' \cos^2 Decl. \sin. 2 Hor. <$, and giving a tide every twelve hours, which varies in magnitude as the square of the cosine of the declination varies, increasing and diminishing twice a year, being also proportional to the cosine of the latitude of the place, and disappearing for a sea situated at the pole. The second part is a diurnal tide, proportional to the sine of the latitude of the given canal, being greatest when the luminary is farthest from the equinox, and vanishing when its declination vanishes.

From these general principles, an attentive student may easily trace for himself the agreement of the theory here explained, with the various modifications of the tides as they are actually observed. It however remains for us to inquire more particularly into the cause of the hitherto unintelligible fact, that the maximum of the spring tides in the most exposed situations is at least half a day, if not a whole day, later than the maximum of the moving forces.

Now it is easy to perceive, that since the resistance observing the lunar period is more considerable than that which affects the solar tide, the lunar tide will be more retarded or accelerated than the solar; retarded when the oscillation is direct, or when $G^2 - B$ is positive, and accelerated when it is inverted, or when that quantity is negative; and that in order to obtain the perfect coincidence of the respective high waters, the moon must be farther from the meridian of the place than the sun; so that the greatest direct tides ought to happen a little before the syzygies, and the greatest inverted tides a little after; and from this consideration, as well as from some others, it seems probable that the primitive tides which affect most of our harbours are rather inverted than direct.

If we wish to apply this theory with precision to the actual state of the solar and lunar motions, we must determine the value of the co-efficients, from the tables of these luminaries. And, first, making the unit of time a whole solar day, in which the horary angle $t$ extends from $0^\circ$ to $360^\circ$, the sun's mean longitude $\Theta$ will be $\frac{t}{365.254}$ added to the longitude at the given epoch, and the moon's approxi- mate horary angle \( t \) will be found from the variation, or the moon's age in space.

Now, in Burckhardt's Tables, p. 57, we find the variation for the midnight ending 1823, by adding the constant quantity \( 9^\circ \) to the epoch for 1824, and \( (11^\circ 14' 44'' 44'') + 9^\circ = 11^\circ 23' 44'' 44'' \), or \( (6^\circ 15' 16'') \), according to the time of Paris. The movement for 12 hours is \( 6^\circ 5' 43'' \); consequently at noon, or 1824 Jan. 1st, astronomical time at Paris, the variation is \( -9^\circ 33'' \), corresponding to the movement of \( 18^\circ 49'' \) in mean time, and the mean conjunction will take place at \( 18^\circ 49'' \) Parisian time, which may be more compendiously expressed by calling it the true mean noon, in the time of the island of Guernsey or of Dorchester; and the movement in 24 hours being \( 12^\circ 11' 26'' \), we shall have \( t = 360^\circ - 12^\circ 19'' = 347^\circ 81'' \) when \( t = 360^\circ \), or \( t = \frac{347^\circ 81''}{360} = 0.96614 \); and the moon's horary angle, considered in relation to the circumference as unity, will always be \( 0.96614 \), \( t \) being the number of days elapsed from the noon of 1st January 1814 at Guernsey.

The sun's mean longitude for the same epoch is \( (279^\circ 35' 23'' 19'') = 77666 \); his longitude for any other time will therefore be \( 77666 + 0.027386 \times t \), and that of the moon \( \varphi = 77666 + 0.03386t \).

We may compute, with sufficient accuracy, the effect of the modifications produced by the change of the moon's distance, or the inequality of her motion in her orbit, or of the periodical change of the inclination of her orbit to the equator, which takes place from the revolution of the nodes, by simply considering the changes which will be produced in the forces concerned by these inequalities, and supposing the effects simply proportional to their causes. If, however, it were desired to determine these modifications with still greater precision, we might deduce approximate formulas for expressing them from the elements employed in the Tables.

The epoch of the moon's mean anomaly for 1824 is \( (4^\circ 29' 25'' 23'') + 2^\circ = 151^\circ 25' 23'' \); the movement for 12 hours \( 18'' \) is \( (6^\circ 31' 57'') + (9^\circ 47' 9'') + 27'' = 6^\circ 42' 12'' \), which gives \( 158^\circ 7' 35'' \) for the mean anomaly at noon in the island of Guernsey. The daily movement being \( 13^\circ 3' 9'' = 13.065^\circ \), the mean anomaly will always be \( 158^\circ 127'' + 13.065t \), reckoning \( t \) from the supposed epoch or day. The principal part of the central equation will then be, according to Burckhardt, \( 262924'' \sin Anm \), or \( (6^\circ 19' 2'') \sin (158^\circ 127'' + 13.065t) \), and its sine will be very nearly \( 11 \sin (13.065t + 158^\circ 127'') \), which will represent the principal inequality of the longitude and of the variation, so that the variation, instead of \( 12^\circ 19' \), will become \( 12^\circ 19' + 63^\circ 3'' \sin (13.065t + 158^\circ 127'') \), and this subtracted from \( 360^\circ \), leaves \( 347^\circ 81'' - 63^\circ 3'' \sin (13.065t + 158^\circ 127'') \), the sine of which is nearly \( 347^\circ 81'' - \cos 347^\circ 81'' \cdot 11 \sin (13.065t + 158^\circ 127'') \).

The equatorial parallax is nearly \( 57'' + 187'' \cos Anm \), or \( 57'' + 31' \cos (13.065t + 158^\circ 127'') \); and the disturbing force, which varies as the cube of the parallax, or of \( 37'' [1 + 0.544 \cos (13.065t + 158^\circ 127'')] \), may be expressed, with sufficient accuracy, by \( 1 + 1.632 \cos (13.065t + 158^\circ 127'') \).

The supplement of the node for 1824 is \( (2^\circ 10' 56'') + 2^\circ = 70^\circ 58' \), to which we must add \( (3^\circ 10' 6'') t \) for the time elapsed; and the longitude \( \varphi \) will be \( 279^\circ 35' 23'' 19' + (13^\circ 10' 35'') t \).

Although the value of the co-efficient \( B \) is not directly discoverable, we may still obtain a tolerable estimate of its magnitude in particular cases, by inquiring into the consequences of assigning to it several different values, equal, for example, to the co-efficient of the solar or lunar tide, or greater or less than either; while we assume, also, for the co-efficient of the resistance, \( A \), a great and a smaller value, for instance \( \frac{1}{3} \) and \( \frac{1}{10} \), supposing \( D \) to be inconsiderable.

We then find, from the expression \( \sqrt{(a^2 + b^2)} M = \sqrt{(a^2 + b^2)} B \) (Theorem K, Schol. 2),

\[ B(S,M) = \frac{\sqrt{(GG - B)^2 + AAGG}}{A} \]

for the solar tide, \( G \) being

\( B = \frac{1}{3}, \quad 0.93442, \quad 1, \quad \text{or} \quad 4; \)

\( A = \left\{ \begin{array}{ll} \frac{1}{3}; & -980, \quad -7550, \quad 10, \quad 13324; \\ \frac{1}{2}; & -832, \quad -2742, \quad 3, \quad 13352; \\ \end{array} \right. \)

and for the lunar, \( G \) being \( 95614 \), and

\( A = \left\{ \begin{array}{ll} \frac{1}{3}; & -1122, \quad 10, \quad 8197, \quad 13036; \\ \frac{1}{2}; & -913, \quad 3, \quad 2942, \quad 12988; \\ \end{array} \right. \)

respectively.

Hence it appears, that the resistance tends greatly to diminish the variation in the magnitude of the tides, dependent on their near approach to the period of spontaneous oscillation, and the more as the resistance is the more considerable; and supposing, with Laplace, that in the port of Brest, or elsewhere, the comparative magnitude of the tides is altered from the proportion of 5 to 2, which is that of the forces, to the proportion of 3 to 1, the multipliers of the solar and lunar tides being to each other as 5 to 6, we have

\[ \frac{36BB}{(1 - B)^2 + A^2} = \frac{25BB}{(n - D)^2 + A^2} \]

whence we find that \( B \) must be either \( 0.9380 \) or \( 0.6328 \); and the former value making the lunar tide only inverse, we must suppose the latter nearer the truth; and the magnitude of the tides will become \( 1.663 \) and \( 1.998 \). And it appears from the same equations, that, \( n \) remaining \( = 0.9342 \), \( A \) cannot be greater than \( 0.6328 \), and \( B \) would then be \( 0.78540 \); and if \( A = 0 \), the values of \( B \) would be \( 0.9617 \) or \( 0.6091 \). It seems probable, however, that the primitive tides must be in a somewhat greater ratio than this of 2 to 1 and 5 to 3, when compared with the oscillations of the spheroid of equilibrium; and if we supposed \( B = 0 \) and \( A = 0 \), we should have \( 7.071 \) and \( 9.756 \) for their magnitude. Now if \( B = 0.6328 \), the tangents of the angular measures of the displacement,

\[ \frac{B}{a} = \frac{AG}{GG - B} \]

become \( \frac{1}{0.96614} \) and \( \frac{1}{0.30160} \)

respectively, giving us \( 69^\circ 50' \) and \( 72^\circ 40' \) for the angles themselves; and if \( B = 0 \), these angles become \( 45^\circ \) and \( 70^\circ 24' \) respectively; the difference in the former case \( 25^\circ 50' \), and in the latter \( 25^\circ 24' \), which corresponds to a motion of more than twenty-four hours of the moon in her orbit.

It appears, then, that, for this simple reason only, if the supposed data were correct, the highest spring tides ought to be a day later than the conjunction and opposition of the luminaries; so that this consideration requires to be combined with that of the effect of a resistance proportional to the square of the velocity, which has already been shown to afford a more general explanation of the same phenomenon. There is indeed little doubt, that if we were provided with a sufficiently correct series of minutely accurate observations on the tides, made, not merely with a view to the times of low and high water, but rather to the heights at the intermediate times, we might by degrees, with the assistance of the theory contained in this article, form almost as perfect a set of tables for the motions of the ocean as we have already obtained for those of the celestial bodies, which are the more immediate objects of the attention of the practical astronomer. There is some reason to hope that a system of such observations will speedily be set on foot by a public authority; and it will be necessary, in pursuing the calculation, on the other hand, to extend the formula for the forces to the case of a sea performing its principal oscillation in a direction ob- For such a sea, the calculations would be somewhat complicated, except in the case of its being situated at or near the equator. We should then obtain, by proper reduction, for the volume of the force, putting \( D \) the sine of the duration, or of the angle formed by the length of the canal with the equator, and \( d' \) its cosine, the expression \( D \sin \cos \text{ Decl. cos. Hor.} < + d' \cos^2 \text{ Decl. sin. cos. Hor.} < \); and the order of the phenomena would be less affected by the alteration of the situation of the canal than could easily have been supposed, without entering into the computation. This expression, when \( D = 0 \), becomes, as it ought to do, identical with the former, making \( L = 0 \).

The preceding theoretical part of this article is that which the late Dr Thomas Young contributed to the Supplement to the former edition of this work, nearly at the time when the concluding volume of the Mécanique Céleste appeared, treating on the same subject. But we presume that, on a close examination, Dr Young's mode of investigation will be found to be of a far more elementary and intelligible description than that of Laplace; and not only so, but it accords much more closely with the phenomena, without requiring the aid of certain very questionable assumptions, which are indispensable in the method followed by this illustrious foreigner. Indeed, under so many different aspects had Dr Young at one time or another studied the subject, that even in his earlier writings are to be found some of the speculations on the tides, which have recently been advanced as entirely new. But in his time, as he was well aware, the real facts recorded and published were so scanty, and the rest so uncertain, that he had no proper data on which to speculate with confidence.

Considering how important it must be for the purposes of navigation and commerce that the tides should be predicted with some degree of certainty, it might have been expected, that when once Newton had furnished a satisfactory mode of explaining in a general way the phenomena of the tides, attention would immediately have been directed, in the maritime parts of all civilized countries, to obtain if possible such an acquaintance with the actual state of the tides as might, with the aid of theory, have served for predicting them, at least approximately. So far, however, were Newton's speculations from being immediately followed up by extensive and accurate observations in every trading port, that during a whole century which succeeded to his time, the subject had been most unaccountably neglected, observations having been made in but very few places, and those only during portions of that long period; nor had any proper efforts been made to deduce from them so much as the empirical laws of the tides. Thus it was only in 1829 that the very general interest which is now taken in the subject may be said to have commenced, with Mr Lubbock's examination of the tide observations which are regularly made and recorded at the London docks. The discussion of these he undertook with the view of obtaining correct tables for predicting the time and height of the tides for the British Almanac. When, however, the principal inequality of the tides, the half-monthly or semimensual inequality, had been determined with sufficient accuracy for practical purposes, the further researches in this difficult problem could not have been so soon undertaken, but for the interest felt in the subject by some other distinguished men of science, particularly by Mr Whewell, and for the pecuniary assistance afforded by the British Association.

The publication of the above-mentioned researches of Mr Lubbock in the Philosophical Transactions for 1831, drew the attention of mathematicians as well as of navigators to the subject of the tides; but it was Mr Whewell who aroused more general interest, and, assisted by the Admiralty, engaged the co-operation of observers in various parts of the world. The subject is well deserving the attention and assistance of every enlightened maritime and commercial nation; for, to do it justice, the study of the tides ought to be pursued in the same manner as that of the other provinces of astronomy; that is, constant and careful observations of the phenomena should be made, reduced, and discussed, at the public expense, so as both to test the accuracy of the tables already framed, and to supply the means of rendering them still more accurate. In this manner also any new corrections, and any changes in the elements of the old corrections, would be brought into view as soon as there were evidence of their existence. Till the problem of the tides is thus treated in a manner worthy of its scientific importance, and of the promise which it now holds forth, it must be regarded as a blot in that system of the national cultivation of astronomy, of which our public observatories afford, in other departments of that science, such effective and magnificent examples.

Mathematicians have not yet succeeded in referring the phenomena of the tides to mechanical principles by rigorous reasoning; and, considering the difficulties of the subject, there is reason to suppose it will be some time before this problem can be fully solved. In the mean while we have an intermediate type of comparison in the equilibrium theory of Daniel Bernoulli, as given in his treatise Du Flux et Reflux de la Mer; for, by modifying the epochs and other elements which enter into the formulae furnished by that theory, these may be made to represent in an approximate manner the laws of the phenomena.

The equilibrium theory supposes, that if the earth were a perfect sphere completely covered by water, this fluid would assume the same form at any given instant as it would do if the forces then acting upon each particle were invariable in magnitude and direction. Although our globe, being only partially covered by water, does not admit of this system accurately taking place, the distribution of the land is such as to allow a closer approximation to it in the southern hemisphere than in northern latitudes and on our coasts; so that if we further suppose the tide-wave nearly to follow such a law at the Cape of Good Hope, or in some still more southern region, and that it is thence propagated northward along the Atlantic Ocean, and round our island preserving all the while a certain proportion of the magnitude and velocity which it had when first formed under the action of the sun and moon; then, upon these suppositions, which are virtually those of Bernoulli, formulae may be framed which will admit of being adjusted by comparison with observation, so as to express the variations in the time and height of high water at any given place, if the time in which the tide-wave is propagated do not vary. But although to a certain extent this equilibrium theory seems thus to suggest and express the laws of the various inequalities of the tides, it must not be rated above its value. It is not the true theory, but a very inaccurate and insufficient substitute for it, which we are compelled to adopt in the present imperfect state of the science of hydrodynamics. The tides are a problem of the motion, not of the equilibrium, of fluids; and we can never fully explain the circumstances of the phenomena till the problem has been solved in its genuine form. Mr Whewell however thinks it is not too much to expect that it may hereafter be rigorously shown from mechanical principles, that the form of an irregular fluid mass constantly dragged along by certain forces, shall at every instant resemble the form of equilibrium which the forces would produce at some anterior epoch, the anterior epoch being somewhat different for the different features of the fluid form. If such a hydrodynamical proposition could be established, almost all the facts hitherto discovered respecting the tides would be fully explained. In the able article by Mr Lubbock, we have the first accurate comparison of Bernoulli's theory of the tides, with the results of observation as deduced from a period of nineteen years in the port of London. The results are important, not merely as furnishing materials and general rules for constructing tide-tables, but also for their general accordance with the theory in question, particularly as regards the semimensual inequality. This agreement was the more important, as affording the indication of a sort of physical connection between that theory and observation, and consequently as justifying such a further examination of its consequences as might lead to the discovery or suggestion of such modifications as would conduce to its general accordance with the laws of all the facts observed. In a subsequent discussion of the tides of Liverpool, in the Phil. Trans. for 1835 and 1836, Mr Lubbock found further evidence in favour of the sort of law to which we have already alluded, and which had indeed been previously suggested by Mr Whewell in his papers on the tides of London and Liverpool; that by referring the tide, not to the lunar transit immediately preceding, but to a transit anterior to the tide by one, two, or more days, the formulae furnished by the equilibrium theory of Bernoulli would be brought into a very near accordance with the observed inequalities in the heights and times of the tides which are due to the changes in the moon's parallax. This was in some respects an important step, particularly as affording convenient expressions for the corrections, since it has been found to apply to a considerable extent to all the periodical inequalities, and seems to be in conformity with the idea that the tides are primarily generated in the Southern Ocean, and thence reach our coasts in the derivative form only. But in one respect it is not satisfactory; for even at the same place very different anterior epochs are required for the several inequalities. Thus, though each formula furnished by the theory can be separately adjusted by trial to represent generally the results of observation for one particular correction at any assigned station, yet since, in the several expressions, the same symbol requires a different value for almost every different correction, such formulæ cannot consistently represent the laws of nature; so that Bernoulli's theory, even when thus modified, is quite incompetent to assign any physico-mathematical reasons for the adjustments in question. The complete solution of the problem would therefore probably require a far more thorough knowledge of the laws of hydrodynamics.

Mr Whewell's researches, on which he has now published a dozen papers, have been chiefly directed to the three following points; first, the motion of the tide-wave at different parts of the ocean; secondly, the comparison of the observed laws at certain places with the theory; and, lastly, the laws of the diurnal inequality of the tide, which seem the most intricate of all. His first memoir, entitled an "Essay towards a first Approximation to a Map of Cotidal Lines," appeared in the Phil. Trans. for 1838. By a cotidal line is meant such as may be drawn through all those points of the ocean which have high water simultaneously. The cotidal line for any hour may be considered as representing the summit or ridge of the tide-wave at that time; meaning by the tide-wave, that protuberance of water upon the surface of the ocean which moves along the seas, and by its motion brings high water and low water to any place at the time when the elevated and the depressed parts of the watery surface reach that place. The cotidal lines for successive hours represent the successive positions of the summit of this wave; so that if a spectator were detached from the earth to perceive the summit of the wave, he would see it travelling round the earth in the Southern Ocean once in twenty-four hours, accompanied by another at twelve hours distance from it, and both sending branches into the narrower seas; and the manner and velocity of all these motions will be assigned by means of a map of cotidal lines. By analysing the movements of the tides according to the most simple considerations of the laws of fluid motion in open seas and in channels, and by explaining the circumstances of their convergence or divergence, their interference with each other, their retardation in shallow water, and their consequent tendency to sweep around the coasts, and to approach them almost perpendicularly; and further, by discussing very carefully all the materials with which nautical surveys and books of navigation could furnish him, Mr Whewell was enabled to construct a map, which not only represented the general circumstances of the tides of the coasts of Britain, but likewise, as he supposed, of the movement of the great tidal wave from the Southern Ocean to the coasts of Europe through the Atlantic; as also its progress in the Indian Seas, and on the coasts of New Zealand.

In order to correct his first approximation to a map of cotidal lines, the British Association, at the instance of Mr Whewell, procured a very extensive series of observations to be made on the coasts of Britain and Ireland, at 537 stations of the coast-guard, in June 1834. These were not only repeated in June 1835, but simultaneous observations were also made, at the request of our government, by the other great maritime powers of Europe and North America. The chain of places of observation extended along the coast of North America, from the mouth of the Mississippi, round the keys of Florida, and northward as far as Nova Scotia; and along the shores of Europe, from the Straits of Gibraltar to the North Cape of Norway. The number of stations was, twenty-eight in America, seven in Spain, seven in Portugal, sixteen in France, five in Belgium, eighteen in the Netherlands, twenty-four in Denmark, and twenty-four in Norway; and observations were made by the coast-guard of this country at 318 places in Britain, and at 219 in Ireland. The observers were directed to record the times of high and of low water, and the height of the surface at each of these times, measured from a fixed point. At each place, the differences between the time of high water, and the time of a preceding transit of the moon, which differences Mr Whewell calls the lunotidal intervals, were taken for the whole series of observations. The immense mass of observations thus furnished was reduced under Mr Whewell's directions; and some of the results, which are extremely important and interesting, have been published in the Phil. Trans. for 1835 and 1836. The last of these publications is accompanied by a second map of the cotidal lines of the coasts of Europe, and by a chart, which shows, by means of a peculiar notation, the range, in yards, of the spring tides at the different stations of observation.

Among other remarkable conclusions which have resulted from these observations, may be mentioned the rotary motion of the tide-wave, which, entering the German Ocean between the Orkneys and Norway, sends a southerly detachment along the coasts of Britain, which is reflected from the projecting coast of Norfolk upon the north coast of Germany, and meets the tide-wave again on the coast of Denmark. Owing to this interference of different tide-waves, the tides are almost entirely obliterated on the coast of Jutland, where their place is supplied by nearly perpetual high water. Indeed this must always be the case wherever one tide-wave continually arrives about six hours later than another, or where more tide-waves successively arrive at still shorter intervals.

Mr Whewell's second object was to compare the observed laws of the tides with the theory, or to propose such modifications of the forms of the theory as would reconcile it with observations. In his very ingenious memoir "On the Empirical Laws of the Port of London," he attempts to deduce from observation, and from very simple considera- Tides.

The character of the formulae for determining the establishment, the semimensual inequality, the corrections for lunar and solar parallax and declination, both as affecting the times and the height of high water. His papers on the "Empirical Laws of the Tides of the Port of Liverpool," and on the "Solar Inequality and Diurnal Inequality" of the tides at the same place, are full of valuable suggestions, which the subsequent investigations of Mr Lubbock have in some cases very remarkably confirmed and extended. Several of the last of Mr Whewell's papers relate to the constancy of the half-tide level, and the diurnal inequality of the heights and intervals of the times of the tide. The discussion of the Liverpool observations had exhibited this last, though under circumstances much less striking than those which characterize its appearance at other places, as will be noticed more particularly afterwards. His first memoir on this subject relates to the diurnal inequality at Plymouth and Singapore; at the last of which places its magnitude is very remarkable, making a difference of not less than six feet in the height of the morning and evening tides, and quite sufficient to obliterate, under certain circumstances, one of the semidurnal tides, and explaining certain peculiarities in the phenomena which have hitherto been considered as cases of interference.

The researches of Mr Lubbock and Mr Whewell on the tides, and their discussions of the observations made at various ports, have now reached such an extent as to be distributed through ten successive volumes of the Phil. Trans. (1831-1840). Some shorter articles have also been published in the Reports of the British Association, and abstracts in the Philosophical Magazine and other journals. Mr Lubbock has besides published a small Elementary Treatise on the Tides. Before these gentlemen took up the subject, which they have done with great credit to themselves and advantage to the public, the theory of the tides, though little cultivated and little known, was in advance of observation. Tide-tables were constructed by unpublished rules, which, though generally very imperfect, formed a profitable possession to those to whom they were known; and the distinctive characters of the tides in the different ports of this kingdom, that of Liverpool perhaps excepted, were confined to the experience and tact of those who were accustomed to use them; but how different is the case at present. The rules for the construction of tide-tables are not only public property, but are based upon the most extensive observations; laws whose existence was hardly suspected, are now distinctly laid down; the progress of the waves in the most frequented parts of the ocean is beginning to be accurately developed; theory, which formerly was in advance of observation, though greatly improved in those parts of it which do not involve the hydrodynamical laws of the ocean, is now greatly behind it; and such a basis of facts has been laid down as may enable the mathematician to commence such a series of investigations as may terminate in enabling some one to give to the theory of the tides a form more closely resembling, in the certainty of its predictions, the almost perfect theory of physical astronomy.

Mr Lubbock and Mr Whewell have, among other subjects, treated at considerable length on the causes of inaccuracy in tide records and observations. We shall here endeavour to notice the chief of these; and it will serve two purposes if we give a popular account of them, in connection with the principal empirical corrections which these gentlemen have found requisite to compensate for the various inequalities of tidal phenomena. On comparing the times of the tide at different places, as hitherto stated by the best authorities, we find very many cases which seem quite incompatible with any notion of obvious continuity and simple laws prevailing in this class of facts. For instance, if the time of high water at Plymouth were five o'clock, and at Eddystone eight, as usually stated, the water must be falling for three hours on the shore, while it is rising at ten or twelve miles distance, and this through a height of several feet. It is difficult to conceive how any elevation at the one place should not be transferred to the other in a much shorter time. In fact, eight o'clock, instead of being the time of high water at Eddystone, is the time of slack water, or when the current changes. Thus there is reason to think that very many, if not the whole, of such inconsistencies have originated in confounding two different phenomena, namely, the time of high water, and the time of the change from the flow to the ebb current. In some cases the one, and in some cases the other, of these times has been recorded as the hour of the tide. The time of slack water, or of the change of current, only coincides with the time of high water very near the shore, and with its influence, and especially in harbours which have only one opening. The difference between these times is generally considerable, and great confusion has arisen from not properly distinguishing between them. The persuasion, that in waters affected by tides, the water always rises while it runs one way, and falls while it returns in the opposite direction, though very erroneous, has long been quite general. For example, it has been usual to state that the time of tide in the British Channel must be three hours later in the mid sea than near the shore, because the current continues running eastward three hours after the time of high water on the coast. Many instances could be given of the perplexities which have arisen from this assumption.

The times most usually recorded as the tide-hours for different places, are those of the tides on the days of new and full moon; which times are often called the establishments of the places to which they belong, and are supposed to regulate the times of the tide on all other days of the lunation, because the tide is primarily governed by the moon. This, however true as a first approximation, assumes that the tide always occurs at the same hour-angle from the moon. But the hour of the tide on any day expresses its hour-angle from the sun; and as the moon changes her right ascension by about forty-eight minutes every day, the observed hour of the tide being given on the day of full and new moon, the hour-angle from the moon may be different according to the time of the day when the conjunction takes place, compared with the time of the observed tide. Thus, if the conjunction take place at one o'clock in the morning, and the observed tide at eleven at night, the distance of the tide from the sun is eleven hours; but at eleven at night the moon is to the east of the sun by her motion in twenty-two hours, which is forty-four minutes of hour-angle, and therefore the tide is only ten hours sixteen minutes behind the moon. But if the observed tide occur at one in the morning, and the conjunction at eleven at night, the moon, at the time of the tide, is forty-four minutes to the west of the sun, and the tide occurs at one hour forty-four minutes. In the former case the establishment is forty-four minutes less, in the latter forty-four minutes more, than the observation of the hour of the tide gives it. If the observed hour of tide were six in the evening, and the conjunction occurred at one in the morning, the true establishment would be 5h. 26m.; but if the tide be at six in the morning, and the conjunction at eleven at night, the true establishment will be 6h. 34m. Thus, an observation of the hour of the tide on the day of new or full moon, leaves an uncertainty of at least 1h. 8m. as to the establishment, if we do not take into account whether the morning or afternoon tide was observed, and at what hour the conjunction or opposition of the moon took place. Besides, the time of high water may often be doubtful to the extent of ten or fifteen minutes, from inaccuracy in the observation; and as this error may occur in opposite directions at two different observations, and may be combined with that of the observation just mentioned, we may thus have two establishments, differing above In the determination of the establishment little accuracy is to be expected, unless numerous observations be used; in this case, the mean of the morning and evening tides may be taken, the effect of the intervals by which the conjunctions and oppositions of the moon precede and succeed noon being supposed to compensate each other. But when the establishment is to be collected from only a few observations, it will be proper to calculate in each case the initial interval or hour-angle by which the tide is distant from the moon.

The time of high water does not follow the moon's transit at the same interval at every period of the lunation; on the contrary, this lunital interval is sometimes greater and sometimes less than that corresponding to the new and full moon, and is regulated by the distance of the moon from the sun. The difference between this and the mean interval is called the half-monthly or semimensual inequality. When the moon and sun are in conjunction, the corresponding tide follows the moon by its mean interval. When the moon is at various hour-angles after the sun, the following are the mean corrections of the mean interval, negative and positive. But the law and magnitude of such numbers depend on the relative effect of the sun and moon upon the tides: the amount is different at different places, and varies with the declinations and parallaxes of the luminaries.

Hour-angle of the moon: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 hours. Correction of establishment: -0, -16, -31, -41, -44, -31, 0, +31, +44, +41, +31, +16, 0 minutes.

Thus, if the establishment corresponding to the new and full moon be six hours, the time of the high water, when the moon is one hour from the sun, will be 5h 44m after the moon's transit; when the moon is two hours from the sun, the time of tide will be 5h 29m after the transit; and so on. When the moon is six hours from the sun, the corresponding time of high water will again coincide with the mean, after which the lunital interval will exceed the mean, till the next conjunction or opposition, and then the same cycle recurs. Hence if the establishment were collected from any observation of the tide not corresponding to the day of new or full moon, it would be liable to an error. If the establishment were six hours by an observation made when the moon's hour-angle was four hours, and compared with the time of the moon's transit, it would appear to be 16m; but by an observation made when the moon's hour-angle was eight hours, it would appear to be 6h 44m. This difference in the results would be avoided by taking the tide which corresponds to new or full moon, or by applying the proper correction according to the preceding table when any other tide was observed. Some authors make this correction depend on the day of the moon's age; but as this is an inaccurate mode of determining the angular distance of the sun and moon, the proper way is to make the correction depend on the difference of right ascensions of the sun and moon, that is, on the time of the moon's transit expressed in apparent time as above. The chance of error could be removed more effectively by taking the mean of the intervals between tide and transit, during a half lunation, or any whole number of half lunations.

Although, according to theory, the semimensual inequality of the time depends on the ratio of the force of the moon to that of the sun, yet the observations show it to be different at different places. The total amount of this inequality, that is, the difference between the greatest and least lunital intervals, is 95 minutes at Plymouth and 93 at Bristol, which exceed its values at many other places. It is 90 minutes at London and Sheerness, 86 at Liverpool and Howth, 84 at Leith, 83 at Portsmouth and Pembroke, 81 at Ramsgate, and only 80 at Brest. Each of these is determined from observations so numerous as to be certain within a minute or two. We see, therefore, how different the mass of the moon would be found to be when deduced from the observations of different places; so that the near agreement of the mass deduced by Laplace from the tides of Brest, with the mass obtained from other phenomena, must have been merely accidental; and this also puts an end to all attempts to correct the tables of the tides by means of the mass of the moon.

The semimensual inequality of the height of the tides, when compared at different places, gives a similar result. It is not, however, the actual amount of this inequality, but a proportion to the mean tide, that is to be taken. At Portsmouth, the mean range of the tide is 12.5 feet; the total semimensual inequality, or difference between the high waters at neap and spring tides, is 2.5 feet; that is, only 1/4th of the mean tide. But at Plymouth, where the mean tide is also 12.5 feet, the total semimensual inequality is 3.4 feet, and thus the fraction is \( \frac{1}{3.4} \). At Bristol, again, where the mean range of the tide is 33 feet, the semimensual inequality is 10 feet, or \( \frac{1}{3.3} \) of the mean tide.

In tide records, as formerly kept, it is often uncertain whether the height marked, mean that of the stream tide, or only the high water on the days of new or full moon.

From the general observations made simultaneously in Europe and America in June 1835, it appears that the amount of the semimensual inequality of the tide is very different at different parts of the world; and though these observations were of too rude a kind to give the amount of the difference, they are sufficient to prove its existence; especially when coupled with a reason for the difference, namely, that the spring tides being higher than the neaps, the tides of the two kinds may travel with velocities which at different places have different relations. Mr Whewell conceives that the observations afford evidence of a local as well as a general semimensual inequality. The changes of this inequality are not obviously explicable. On the coast of North America, the amount of the difference of the greatest and least lunital intervals is small, being generally less than 80 minutes, and at Newport as low as 56. On the coast of Portugal, at several places, this difference is so small as almost to throw doubt on the accuracy of the observations. At Pera in Algarve it is only 42, at Lagos Bay 24, while at Peniche it is 130. On the greater part of the French coast it ranges with great steadiness from 80 to 100, except at the little harbour of Abrevraak, where it is 125 minutes. At Torr Head, on the north coast of Ireland, it is 146, and at Rachlin Island 240; but these are cases of extreme irregularity. On many parts of the south coast of England it is small, being from 70 to 74, as at Exmouth, Weymouth, St Albans Head, St Lawrence, Swanage Bay, Brighton, and Hastings.

From the same observations, it appears that the amount of the semimensual inequality of the height also varies. In general, the greatest range is twice or twice and a half the smallest; but this is so far from being a universal rule, that many of the cases which at first seem to agree with it, really differ widely when allowance is made for the diurnal inequality. Thus, at Mount Desert Island, near the coast of America, the whole amount of the semimensual inequality of high water is about three feet in a tide of thirteen feet, thus reducing the smallest range to eleven; but the diurnal inequality reduces it further to eight feet.

If, as is generally believed, the most of the tides be ge- generated in the Southern Ocean, the tide which is produced under the more immediate influence of the new or full moon cannot, at most places, be that which takes place on the day of new or full moon; and in reading the preceding sketch of the phenomena, these must not be confounded. The latter, however, is that which has commonly been observed for the purpose of determining the establishment; but it often differs materially from the former, and in the tide-records anomalies frequently occur which seem referable to this difference. It has already been mentioned, that the agreement between the empirical laws of the tides and the equilibrium theory is much improved, by referring the tides to a transit anterior to their occurrence by two or three days. To illustrate this, Mr Lubbock denotes the successive transits of the moon (at intervals of about twelve hours) by the letters A, B, C, D, E, F; F being the transit which immediately precedes, by about two hours, the high water at London: and he finds that the laws of the tides at London and Liverpool agree best with theory when they are referred to the transit B. The tide which reaches London at two hours after the transit F, was at Plymouth about six hours after the transit E; and as from transit B to transit E is about thirty-eight hours, if we refer the Plymouth tides to B, we take a transit about forty-four hours before the tide. Transits A and C are about fifty-six and thirty-two hours anterior to the tide. Nearly the same may be said of Bristol. The general result is, that the transit B gives the best tables.

We cannot, however, in this way completely reconcile observation and theory, even as regards the semimensual inequality. From Mr Lubbock's researches on the Liverpool tides (Phil. Trans. 1836, p. 57), it appears, that while the transit A very nearly reconciles the theoretical and observed times, we must take a still earlier transit if we would obtain the like agreement in the heights. Nor does that selection of a transit which best represents the semimensual inequality bring about an agreement with theory in the parallax and declination-corrections. Hence, although there appear to be in the actual laws of the tides inequalities corresponding to all those which arise from the supposition of the equilibrium tide of an anterior epoch transmitted along the ocean to our shores, we cannot so assume the epochs to produce all the inequalities at once even for the same place. The epoch is of one value for the times, of another for the heights; different for the parallax correction, and again for the effect of declination.

The relation between a change in the epoch and the semimensual inequalities may be examined as follows. The successive lunar transits occur at intervals of about twelve hours twenty-four minutes of solar time. The semimensual inequality, both of interval and of height, is referred to the solar time of the moon's transit. The height is the same whether it be referred to the transit E immediately preceding, or to the transit B a day and a half sooner. If the moon moved uniformly in her orbit, the inequality of the interval between the tide and transit would be the same, whether the tide were referred to the transit E, or any anterior one, as B; for the interval is increased by the constant quantity twelve hours twenty-four minutes nearly for every transit that we go back. But though the inequality of the interval for any given tide would be the same, it would not occupy the same place in the table or curve, since it would be referred to a different hour of the transit. For example, if the mean interval of transit and tide at Bristol referred to E be 7h., and if, when the moon's transit is 2h., the tide be at 8h., the interval is 6h., and therefore in this case the inequality is minus one hour. If now we refer this tide to the transit B a day and a half sooner, the interval of the transit and tide will be 6h. + 36m. + 72m., and the mean interval will be 7h. + 36m. + 72m.; and therefore, as before, the inequality is minus one hour. But the moon's transit F taking place at 2h. solar time, the transits D, C, B will take place at 1h. 36m., 1h. 12m., 0h. 48m. respectively; and therefore the inequality of minus one hour, which was referred to the transit happening at 2h. when transit E was used, is referred to the transit at 0h. 48m. when we employ the anterior epoch B. Thus, by referring to an anterior epoch, the whole semimensual inequality is shifted backward through twenty-four minutes of lunar transit, for every step of one transit backwards. This is the mean result, supposing the moon's motion uniform, and neglecting all other inequalities; on which suppositions nothing would be gained or lost in accuracy by the change of the epoch. But in the actual case, this mean result is modified by the influence of the other inequalities, which make one transit a better epoch than another.

The tide which comes to the shores of narrow and long seas is not immediately produced by the luminaries, but is derived from the tide in the main ocean. Its circumstances are governed by those of the primary tide from which it is derived; and whatever interval may elapse during its transfer, it is regulated by the position which the sun and moon had at the time when they determined the primary tide. Now this time may have been one, two, or more days before the tide reaches the place where it is observed. Thus the tide on the shores of North America and Spain, conformably with its being supposed to be generated in the Southern Ocean, and sent up the Atlantic, seems to correspond to the configuration of the sun and moon at a day and a half previous. The tide in the port of London appears to be two days and a half old when it arrives. The time spent in the transfer of the tide is very different in different parts of the world, and has not yet been ascertained for many places. But it affects the determination of the establishment from observations, in a manner which may be thus explained. If the tide at London be determined by the position of the sun and moon two and a half days before it occurs, the moon must then have been more to the west of the sun by an angle of two hours (her motion in AR in two and a half days) than she is when the tide arrives. Hence the tide which happens on the day of full moon, corresponds to the period when the moon was in AR, two hours west of the point opposite to the sun, or ten hours east of the sun. Therefore, by the preceding small table, the tide is thirty-one minutes later than the mean interval between the tide and moon's transit. The tide at London takes place nearly at two o'clock on the days of new and full moon; and therefore 1h. 29m. is the corrected establishment for London.

The establishment is usually defined to mean the hour of high water at new and full moon. Mr Whewell calls this the vulgar establishment. Observations of tides have most commonly been made with the view of determining this vulgar establishment, which it now appears is not a corresponding quantity at different places. The mean of all the lunidial intervals for a half luation, he calls the corrected establishment. Hence the vulgar establishment seems to be greater than the other by a quantity depending on what Mr Whewell calls the age of the tide; namely, the length of time which has elapsed since its real or theoretical origin. This is also what Mr Lubbock, following Bernoulli, calls the retard. It is considered as the principal reason why the greatest tides do not occur on the days of new and full moon, nor the least at the quadratures, but one, two, or more days after. Both Mr Lubbock and Mr Whewell frequently use the term epoch in the same sense as the age of the tide, or at least as the time during which any particular inequality has been transferred from the Southern Ocean to the place of observation.

The doctrine of the age of the tide, as thus laid down, not unfrequently leads to inconsistencies, some of which are only apparent, being referable to the mixture and inter- ference of different tides; but another reason for some of them, we presume, is, that the luminaries, after generating the primary tides, are not, as they are usually assumed to be, mere idle spectators of the derivative tides, but often continue to act upon them in various ways, accelerating some and retarding others.

Although the observations make the parallax corrections both of height and time vary as the parallax, yet some of the observations do not make the height of the tide itself proportional to the cube of the parallax, as theory requires, but make it more nearly proportional to the square of the parallax, being as the $2^2$ power. The declination corrections both of height and time are found by observation to vary as the square of the sine of the declination, which, according to theory, they should do.

But the difference between the tidal phenomena of the morning and evening of the same day, called the diurnal inequality, is often so considerable, and at some places so remarkable, that it is far less perfectly understood than any of the other inequalities, and furnishes, as we shall afterwards see, a source of far greater uncertainty in the determination of the establishment than any yet mentioned. It had not till of late been introduced, and still very sparingly, into tide-tables. To it, or rather to a want of the knowledge of it, are no doubt owing many of the uncertainties or ambiguities and errors which occur in recorded tide observations. It is of great importance both to the theory of the tides and to the purposes of navigation, that this diurnal inequality should be fully analysed. Mr Whewell's researches have opened a wide field, but here we can only notice them in a very general manner.

Were the earth wholly covered by water, the diurnal inequality of the heights of high and low water would depend on the semidurnal transits of the poles of the equilibrium spheroid being alternately north and south of the equator. For example, if the moon had $20^\circ$ north declination, the tide spheroid would have one pole in lat. $20^\circ$ north, and the other in $20^\circ$ south; and while the earth revolved, a place in lat. $50^\circ$ north would have the tide which belongs to these two poles alternately; and being $30^\circ$ from the one pole and $70^\circ$ from the other, the two tides would be very unequal. On the same principles, in northern latitudes the tide which belongs to an upper transit of the moon should be the greater (of the two on the same day) when the moon's declination is north; when the moon crosses the equator, the difference of the two tides should vanish; when she has south declination, the tide which belongs to her upper transit should be the smaller. The contrary (as to greater and smaller) should be true of the tide which belongs to the inferior transit. The diurnal inequality may therefore be conceived to arise from a wave oscillating in the direction of the meridian, and of which the maximum height comes to each place once in twenty-four lunar hours; the minimum height arriving, of course, at the intermediate twelve hours. For such oscillations the Atlantic and Pacific both afford the most ample scope. If the time of the maximum height of this wave arriving at any port coincide every day with the time of high water, the alternate high water being at twelve hours interval, will be affected alternately with the greatest and least heights of the diurnal wave; and the intermediate low waters will coincide with the mean height of this wave, and will not be at all affected. In this case there will be a decided diurnal inequality in the height of the high water, but none in that of the low water. In like manner, if the time of the maximum height of the diurnal wave coincide with the time of low water, the height of low water will be marked with a diurnal inequality, while the height of high water will exhibit no such feature. But if the diurnal wave arrive every day at a time intermediate between high and low water, it will raise both the high and the low water which are nearest it, and will depress both the high and the low water which happen in the other half of the day. Hence both the high waters taken separately, and the low waters taken separately, will be marked by a diurnal inequality; and this inequality will be greater for high water or for low water, according as the time of the maximum of the diurnal wave is nearer to the time of high or low water.

There are various places at which it has long been known that there is commonly a difference in the morning and evening tide of the same day. It is stated by Colepress in 1668 (Phil. Trans. vol. iii. p. 633), that at Plymouth the tides, from about the end of March till the end of September, are about a foot higher in the afternoon than in the forenoon, and vice versa the rest of the year. But we shall soon see that this way of expressing the fact, by speaking of evening and morning tides, is quite inaccurate. The other inequalities of the tides having been found to follow the laws of the equilibrium theory (although the constant elements, the magnitudes and epochs, can be determined only by observation), and Mr Whewell having found the diurnal inequality very clearly marked in the Plymouth observations, he attempted to trace its laws by assuming a similar correspondence with the equilibrium theory. The result confirmed the assumption in a striking manner, the age or epoch of this inequality for Plymouth being about four days.

At Singapore the diurnal inequality of low water is of a magnitude which could not have been anticipated, the height of the evening and morning tide often differing six feet; although the whole rise of the mean tide is only seven feet at spring tides, and the difference of mean spring and neap tides not more than two feet. The epoch at that place is about a day and a half. It is a curious circumstance, that the sun affects the low water at Plymouth more than the high water; and that the moon's declination at Singapore affects the low water four times as much as the high water, while at Plymouth it affects it less.

It is easy to conceive the diurnal inequality carried a little farther than it is at Singapore, so that at a certain stage of it the alternate tides would entirely vanish. This is equivalent to supposing the highest low water and the lowest high water to have the same height, of which examples will afterwards be given.

Captain Fitzroy having caused observations be made at King George's Sound, on the south coast of New Holland, every half hour for some days, and for a portion of the time every quarter of an hour, found that on March 7th and 8th 1836 there were two very unequal tides, and that on the 9th and 10th there was only one tide; but a recession and return in the high water, which had been barely perceptible on the 11th, became more and more marked on the 12th, 13th, and 14th, so as again to give two tides each day. Thus at this place it appears to be only at one particular period of the semilunation that we have a single-day tide. That of Tonquin was referred by Newton to the interference of two tides which arrive by different channels. The great diurnal inequality of Singapore, which is in the same seas, appears to be clearly due to the effect of the moon's declination; and the determination of this point, and the circumstances ascertained to occur in the reputed single-day tide of King George's Sound, throw some doubt on the explanation just referred to; nor can this doubt be removed till the tides of those seas have been more fully observed.

That this inequality cannot properly be called an excess of the evening tide at one season of the year, and of the morning tide at another, must now be evident from the circumstance that the high water at Plymouth is on the average five hours after the moon's transit. Suppose the moon to move in the ecliptic, which is her average path when the sun's right ascension is five hours (that is, about... June 7), the tide which follows the moon's transit will follow the sun's transit also as soon as the moon is north of the equator; that is, if the diurnal inequality were regulated by the moon's place on the same day, the afternoon tide would be greatest, and so it would continue till the moon was seven hours after the sun, and the tide following the moon would become the morning tide: but at the same time the moon would pass to the south of the equator, and so the tide following the moon would be the smaller. In this situation, therefore, the evening tide would be the greater during the whole lunaion.

It was only of late that the diurnal inequality of the tides began to be attended to as it deserves; for however imperfectly understood, it is no doubt a regular change, considerable in its amount, and almost universal in its prevalence. It would be easy to adduce many cases in which the safety or loss of a ship has been determined by it. Though the existence of such an inequality in particular places has long been known, its laws have been sadly misunderstood. For example, it has been supposed always to affect the morning and evening tides in opposite ways, which is only an accidental and local expression of its rule. In the Phil. Trans. for 1836, p. 57, Mr Lubbock has described how he obtained it for Liverpool; while Mr Bywater, who has introduced it into his tide-tables for that port, and Mr Bunt into those for Bristol, have also collected it from observations. But the connection of this inequality, as it exists in different parts of the world, was never brought into view till Mr Whewell discussed together the European and American observations of June 1835. The laws which the inequality follows, when thus considered on an extensive scale, appear to be very curious, though they are still but very imperfectly known.

In some cases this inequality is most clearly seen in the heights of high water, in others in the low water, and in some cases it greatly affects the times of both. Mr Whewell exhibited the results of June 1835 in curves, by erecting a series of equidistant ordinates to represent the heights of the successive tides above a fixed point at each place; and these curves generally showed a series of parallel zigzags (the tides being alternately higher and lower); and they were so regular and similar as to prove both the goodness of the observations and the existence of the diurnal inequality. This was so remarkable on the coast of America, that scarcely any exception occurred. Next to this, it was conspicuous, especially during a portion of the observations, in Spain and Portugal; then on the west of France, the coast of Cornwall, and parts of the west of Ireland. In the German Ocean this inequality was obvious, but, owing to the mixture of different tides, it was less steady and regular.

Although the diurnal inequality is generally believed to depend upon the sun or moon's being north or south of the equator, yet, as we shall afterwards see, there are cases which it seems very difficult to explain on this principle. Nor does the sign of this inequality, as was long supposed, depend on the place being north or south of the equator. Its maximum corresponds to, but is not necessarily simultaneous with, the moon's greatest declination; nor does the period of its vanishing everywhere coincide with the time of the moon's passing the equator, but more generally is somewhat later. Between periods corresponding to, or rather at equal intervals after, two such passages, the inequality increases from 0 to a maximum, and decreases to 0 again, alternately. The curves which represent the heights really exhibit such alternate increase and diminution; and the inquiry naturally occurs, after how long a time does the moon's position show its effect in the diurnal inequality? In the case of Liverpool, Mr Whewell has endeavoured to show that this inequality expresses the effect of the forces upon the equilibrium spheroid, as they existed six days previously. But this interval is very different at other places, and its range is curious.

From the general tide observations made in Europe and America in June 1835, the results of which have been given by Mr Whewell (Phil. Trans. for 1836, p. 302), it appeared that the diurnal inequality on the east coast of Scotland was, during that semilunation, irregular, passing over a tide in the middle of the series. This and others of its anomalies on the coasts of the German Ocean, appear to show that the waters in that region are affected by the mixture of different tides. On the east coast of America, the changes of this inequality seem contemporaneous with those of the moon's declination; so that the epoch there is zero. On the coasts of Spain, Portugal, and France, it is successively two and three days, which is quite consistent with this epoch, being four days on the coast of Cornwall and Devonshire.

A circumstance not less remarkable in the progress of the diurnal inequality is, that it is seen much more distinctly and steadily at some places, than at others which are near them; nor does it seem easy to assign any rule which it follows in this respect. It is very marked, and almost universal, on the coast of the United States, and was conspicuous in the observations of June 1835 in Spain and Portugal, the west of France, and part of the west of Ireland; yet at intermediate places it could not be detected. It is large on the east coast of New Holland, as we know from Cook's getting his ship off a reef by means of it; and the north and south coasts of Australia appear to exhibit the extreme case of it. We might suppose it to affect the whole of the Indian Ocean; yet at Keeling Island, in the middle of that ocean, it did not decidedly show itself in Captain Fitzroy's observations from the 2d to the 8th April 1836.

The epoch of the diurnal inequality being different in different parts of the world, is a very curious fact, and not easily reconcilable with the notion of a tide-wave travelling to all shores in succession. In accordance with this view, the tide on the shores of America had been considered as identical with the almost contemporaneous tide on the coasts of Spain and Portugal; nor does it seem easy to imagine the form of the tide-waves, so that this shall not be the case. Yet we find that the tides on these two sides of the Atlantic cannot be identical in all respects; for on the 9th, 10th, and 11th of June 1835, when the diurnal inequality was great in America, it was nothing in the west of Europe; and on the 18th and 19th, when this inequality had vanished in America, it was great in Europe. It would seem as if the tidal phenomena on this side the Atlantic corresponded to an epoch of the equilibrium theory, two or three days later than the same phenomena in America; and different kinds of phenomena do not seem to travel at the same rate. Thus the equilibrium theory, though it may explain the general form of the inequalities, cannot give their epochs and amounts by any possible adjustment of constants. The notion of the progress of the tide-wave from south to north in the Atlantic is still further involved in difficulties by its appearing, from the observations of Sir John Herschel, that at the Cape of Good Hope the diurnal inequality showed itself most clearly on the 17th, 18th, and 19th of June, that is, as late as in Spain and Portugal. The diurnal inequality appears also, but not so generally, in the curves which represent the times; nor is this difference always in the same direction. Thus on the coast of America, at some places, the afternoon tides were later than the mean, and those of the forenoon earlier than the mean, for a great part of June 1835, while at other places the reverse was the case.

From certain tide observations made in the Indian Seas, and examined by Mr Whewell, we select the following as striking features in those of Surat Roads in the Gulf of Cambay, Gogah on the opposite side of the same gulf, and Bassadore in the Island of Kismis at the entrance of the Persian Gulf. In the first two places there is an enormous diurnal inequality of The heights, amounting sometimes to seven or eight feet. It appears that this inequality vanishes and changes its sign about two days after the moon's declination does so. At Bassadore there is a very large diurnal inequality of the times, sometimes exceeding two hours, and sufficient to displace the tides. For instance, about the 23rd November 1834, the tide-hour of the forenoon was greater by almost two hours than that of the afternoon, the former being at 5h. 33m. A.M. and the latter at 3h. 41m. P.M. What makes his anomaly still more remarkable is, that at this place there is little or no diurnal inequality of the heights, which is a most curious tidal feature, in addition to those already remarked in the Indian Seas.

From observations made at Petropavlovsk, in the Bay of Avacha, latitude 53° 1' N., long. 158° 44' E. in 1827 and 1828, by the Russian Admiral Lütke, and transmitted to Mr. Whewell, it appears that the high water there is affected in its time by a very large diurnal inequality, while the height is only slightly affected by an inequality of that kind. The height of the surface was carefully observed every ten minutes, day and night; and when near its maximum, every two minutes. This great care and labour, which would have been superfluous at most places, was quite necessary in this instance; for if the observations had not been thus continued, they would not have enabled Mr. Whewell to detect the very curious laws of the phenomena about to be described. This diurnal inequality of the time reaches the enormous amount of above four hours. Thus the intervals between moon's transit and high water, on October 11 and 12, 1827, were as follows: 5h. 38m., 1h. 33m., 5h. 20m., 6h. 52m.; on June 23 and 24, 1828, they were 7h. 9m., 3h. 9m., 7h. 21m., 2h. 51m.; on October 15 and 16, 1828, they were 5h. 13m., 2h. 0m., 6h. 7m., 2h. 46m.; showing an alternate increase and diminution to the extent above mentioned. The greatest alternate inequalities of the heights of high water during these series of observations, were something more than a foot. But the observations of low water are marked by additional features still more curious. For though in these the diurnal inequality of the times really appears, it is neither so large nor so regular as that of high water; it seldom exceeds an hour. The diurnal inequality of the height of low water, on the other hand, is much larger than that for high water, amounting to three or even four feet; and this in a tide of which the whole range rarely exceeds five feet. The diurnal inequality is supposed to depend principally upon the moon's declination; and its maximum and its disappearance have been found, at most places hitherto examined, to follow at a short interval (one or two days) the maximum and the vanishing of the moon's declination. On examining the Petropavlovsk observations with regard to this point, we find that the greatest and most regular of the diurnal inequalities above noticed (the inequality of the time of high water and of the height of low water) correspond with the moon's declination; so that the epoch or age is zero. On the contrary, not only is the epoch of the other two inequalities, that of the heights at high water and of the times of low water, different from that of the preceding, but they alternate with them, vanishing when the others reach their maxima, and showing their maxima when the others vanish. This is a very perplexing circumstance; for if the diurnal inequality depend upon the moon's declination, it is difficult to conceive how its effect upon the height of high water and the time of low water should be greatest just when the moon is in the equator, and that the reverse should hold in respect to the other two inequalities.

The tides of Petropavlovsk show more clearly than any yet examined, how the diurnal inequality may be so large as to lead to the appearance of only one tide in the twenty-four lunar hours. The heights of the high waters are alternately greater and less; as are also, in a still greater degree, the heights of the low waters. Thus some of the high waters are depressed, and some of the low waters elevated, till there is little vertical difference between the two. On the 17th of June, the rise from low to high water in the afternoon was only eight inches, although the rise in the forenoon had been four feet. In the same manner the fall from high to low water in the afternoon of June 22 was only two inches, and in the forenoon of June 23, only one inch, although the intermediate fall in the afternoon of those days was above four feet. From 6h. to 8h. on the 24th the surface remained stationary. Thus one of the two half-day tides being obliterated, we have only one tide in the twenty-four hours. It is to be recollected, however, that this takes place only at a particular period of each lunation, depending upon the declination of the moon. When, therefore, a traveller meets with such a phenomenon, if he would pursue his tide observations for a few days with assiduity, he would probably find the single-day tide resolve itself into the usual case of two daily tides.

The term establishment ceases to be applicable when the diurnal inequality affects the time of high water to so very large an amount as we have seen it does at Petropavlovsk; because at almost any time of the lunation the interval of moon's transit and high water may vary very much from one tide to another, and this uncertainty deprives the establishment of all utility in such cases, unless we also take into account the diurnal inequality.

If the usual diurnal inequality at any place be the effect of a tide-wave arriving at the shore once a day and superimposed upon the semidiurnal tide-wave, it is natural to ask whether such a mode of representation can be applied to the tides now under consideration. Mr. Whewell, after attempting at considerable length to effect this, acknowledges that he is fairly outdone by the inequalities alternating in their vanishing and maxima, as above described.

The tides at the Isle of Sitkho, latitude 57° 2' N. and longitude 135° 18' W., exhibit a very great diurnal inequality both in the heights and times. The amount of this in the time of high water reaches an hour (30m. positive and 30m. negative); at low water it is somewhat less. The diurnal inequality of the height of high water is two and a half feet; at low water its maximum amounted to five feet; the greatest rise from low to high water being about fifteen feet. In other respects this case is similar to that of Petropavlovsk, except that the tides never become single. The same communication of Admiral Lütke contained an account of the great diurnal inequalities of Port de la Coquille and of Port Lloyd, as also of the establishments of several other places in the North Pacific; but, as already stated, the term establishment becomes extremely vague, and almost meaningless, when applied to seas in which the phenomena of the tides are such as above described.

From the account of Admiral Freycinet's voyage, it appears that at several places in the North Pacific, the tides, as observed in that expedition, exhibit features similar to those here noticed. It has often been lamented that so few tide observations have been made in distant parts of the world; but it is now sufficiently clear, that if, in places such as those now noticed, the observations are not made with great care, and continued for a length of time, they can only serve to mislead.

The Rev. John Williams, who, from observations continued during several years' residence in the islands of the Pacific Ocean, had it in his power to furnish more certain information than could be obtained by any transient visitor, however profound in knowledge or diligent in research, has given a very intelligent account of certain tides in those regions in his "Narrative of Missionary Enterprises in the South Sea Islands," p. 200. It is a fact well known to the missionaries, that the tides in Tahiti and the Society Islands are uniform throughout the year, both as to the time of the ebb and flow, and the height of the rise and fall; it being high water invariably at noon and midnight, and low water at six in the morning and evening. The total range from low to high water seldom exceeds eighteen inches or two feet. Generally once, and frequently twice in the year, a very heavy sea rolls over the reef, and bursts with great violence upon the shore. But the most remarkable feature in this periodically high sea is, that it invariably comes from west and south-west, which is just the opposite of the trade-wind. The eastern sides of the islands are never injured by these periodical inundations. Mr Williams is anxious to call the attention of scientific men to this remarkable phenomenon, which he believes to be restricted to the Tahitian and Society Island groups in the South Pacific, and the Sandwich Islands in the North. But he cannot speak positively respecting the islands eastward of Tahiti; but at all the islands he has visited in the same parallel of longitude to the southward, and in those in the same parallel of latitude westward, the same regularity is not observed, but the tides vary with the moon, both as to the time and magnitude of the tide, which is the case at Rarotonga. He is also anxious to correct the erroneous statements of some scientific visitors, particularly of Kotzebue, who asserts that "every noon the whole year round, at the moment the sun touches the meridian, the water is highest, and falls with the sinking sun till midnight!" According to Captain Beechy, "the tides in all harbours formed by coral reefs are very irregular and uncertain, and are almost wholly dependent upon the sea-breezes. At Oututauna, it is usually low water about six every morning, and high water half an hour after noon. To make this deviation from the ordinary course of nature intelligible, it will be better to consider the harbour as a basin, over the margin of which, after the breeze springs up, the sea beats with considerable violence, and throws a larger supply into it than the narrow channels can carry off in the same time, and consequently during that period the tide rises. As the wind abates, the water subsides; and the nights being generally calm, the water finds its lowest level by the morning."

"This statement," says Mr Williams, "is certainly most incorrect; for not only have I observed for years the undeviating regularity of the tides, but this is so well understood by the natives, that the hours of the day and night are distinguished by terms descriptive of its state: as, for example, instead of asking 'What is the time?' they say 'Where is the tide?' Nor can the tides, as Captain B. observes, be 'wholly dependent upon the sea breeze;' for there are many days during the year when it is perfectly calm, and yet the tide rises and falls with the same regularity as when the trade-winds blow; and we very frequently have higher tides in calms than during the prevalence of the trade-wind; besides which, the tides are equally regular on the westward or leeward side of the islands, which the trade-wind does not reach, as on the eastward, from which point it blows. But the perfect fallacy of Captain Beechy's theory will be still more apparent, if it be recollected that the trade-wind is most powerful from mid-day till about four or five o'clock, during which time the tide is actually ebbing so fast, that the water finds its lowest level by six o'clock in the evening; and that in opposition to the strength of the sea-breeze, Captain Beechy adds, 'that the night being calm, the water finds its lowest level by morning;' whereas the fact is, that the water finds its highest point at midnight when it is perfectly calm. How then can the tides be dependent on the sea-breeze?"

Mr Lubbock thinks some of these extraordinary anomalies may perhaps be accounted for by considering the continent of South America as a dam, preventing the derived tide-wave from flowing freely into the Pacific from the great Southern Ocean.

Owing to the tide-wave which enters the German Ocean by the north coast of Scotland, not reaching the mouth of the Firth of Forth at the same time with the other tide-wave which comes through the Straits of Dover, there are frequently four, and sometimes six tides a day, observed in the upper part of the Firth of Forth. These peculiarities are most conspicuous about the time of spring tides, and gradually decrease till the neaps, when they vanish altogether. For the particulars, we beg to refer to the Reports of the British Association on Waves, by Sir John Robison and Mr Russell. But so long ago as the year 1750, peculiarities had been observed in the tides of the Forth, as described by Wright in the Phil. Trans. for that year.

When a tide-wave arrives at the mouth of a river, it enters in a manner similar to that of a derivative tide into a narrow sea; and though there are generally some features peculiar to each river in the entrance of the tide-wave, it afterwards assumes a direction at right angles to the current, and advances regularly up the stream. If the channel gradually narrows upward like a funnel, the height of the tides will be increased by this convergence. Where the width is nearly uniform, the tide will gradually die away, and still more rapidly if the channel spreads out. In the Amazons the tide continues to ascend for several days against the stream, and is still sensible at 200 leagues from its mouth. Seven or eight tides, with intermediate low waters, follow in continual succession up this mighty river. Something similar takes place in the St Lawrence, the tides reaching 482 miles up the main channel to a point between Montreal and Quebec. Among other plausible reasons to account for the tide not ascending the Mississippi, its crookedness has been assigned as a very sufficient one. But the best of all reasons seems to be, that there is next to no tide in the sea at its mouth; the total rise there being only about eighteen inches. When the tide, on entering a river, is made to rise greatly by the contraction of the channel, the part of the water so affected may be abruptly terminated on the inland side, owing to the depth and quantity of the water on that side not admitting of the surface there being immediately raised by means of transmitted pressure. A tide-wave thus rendered abrupt has a close analogy with the waves which curl over, and break on a shelving shore, and is called a bore. In many places it occurs in such magnitude as to produce great noise by the violent intestine motion of the water. Though it appears to travel with great rapidity, it in fact moves more slowly than the tide-wave under any other circumstances. The bore which enters the Severn is nine feet high; that in the creek of Fundy is said to be still higher. In the Garonne this phenomenon takes place near Bordeaux, and is called there the Mascaret; at Cayenne it is called the Borre. It occurs on a grand scale in the river Amazons, at the junction with the Aravary, having there a face twelve or fifteen feet in height, and producing a noise which may be heard at two leagues distance. The Indians of the neighbourhood call it Pororoca. Ships are often exposed to considerable danger from the bore, especially in the more shallow parts of a river, and near the shore.

The range of the tide, that is, the height of high water above low water, is very different at different places, and is affected by circumstances which it is often very difficult to analyse. It is however clear that the configuration of the coast exercises a very considerable influence upon the amount of this range, according as it makes the tides converge or diverge. Thus the range is very much increased in deep in-bends of the shore, which are open in the direction of the tide-wave, and gradually contract like a funnel, such as the Bristol Channel, the Gulf of Avranches, the Bay of St Malo, and the Bay of Fundy, where sometimes it very considerably exceeds seventy feet, some say 100. On the con-