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 20 feet between the greatest and least heights of the water in one day, and sometimes only 12 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 This article on the Theory of Tides, written for a former edition by the late Dr Thomas Young, has been allowed to stand here in its integrity, mainly from the acknowledged excellence of the article by eminent scientific men of the present day. The more recent inquiries of Lubbock and Whewell into the subject will be found noticed in the Sixth Dissertation, art. 69, &c., and its more practical aspects will be found treated off sufficiently under the head Physical Geography, § 66, &c.—Ed. 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. Cæsar 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 (Somn. Seip. 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, iii. 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. 553), 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 synoptical 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 from that of Laplace, without some direct observation. It might at least be supposed very easy to enumerate the existing observations, scarcity 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 | 2.20W | Brest | 1.24 | Secotera and C. | 20.30W | | Cape of Good Hope | 1.14E | Bayonne | 1.20 | Guardafui | 21.30 | | St. Helena | 0.23W | C. St. Vincent | 5.13 | Ulstein | 21.39 | | Cape Corse | 0.7 | Corunna | 3.35 | Huacheine | 21.53 | | Rio Janeiro | 2.53 | Belleisle | 4.53 | Shoreham | 22.1 | | I. Martin Vaz | 1.56 | Palmiras Pt. | 5.41 | Foual Pt. Mad. | 22.1 | | Ascension | 0.57 | Port Cornwallis | 6.27 | Botany Bay | 22.53 | | Christmas Sound | 4.46 | Rochelle | 7.10 | St Valery en Caux | 22.12 | | St Jago. | 1.34 | Yannes | 13.34 | Macao | 22.24 | | Port Desire | 4.20 | St Paul de Leon | 8.40 | St Valery sur Somme | 22.24 | | St Helena, S. A. | 4.40 | Morlaix | 8.40 | Dunross | 22.20 | | Quibio | 5.29 | Rochfort | 8.59 | Brighton | 22.31 | | Sierra Leone | 0.53 | Bear Island | 9.40 | Brighton | 22.35 | | Easter Island | 7.19 | Christmas Island | 9.19 | Dunross | 22.35 | | St Julian's | 4.32 | Chloe | 9.20 | Beach Head | 22.50 | | Magellan Mouth | 3.20 | Cape Clear | 9.20 | Cowes | 22.50 | | St John's, Newf. | 3.90 | Antigua | 9.20 | Needles | 22.56 | | Guadaloupe | 4.7 | St Peter and Paul | 10.7 | Anholt | 22.13 | | Panama | 5.21 | Awatsha | 7.47 | Boulogne | 22.33 | | Tortugas | 4.51 | Kinsele | 10.51 | Hastings | 22.21 | | Cape Blanco | 8.16 | Eddystone | 10.54 | Deal Castle | 22.36 | | Bermudas | 4.14 | Falmouth | 11.14 | Dover | 22.49 | | Martinique | 4.5 | Rotterdam L. | 11.33 | Dungeness | 22.49 | | Guayaquil | 5.17 | Drake's L. Plymouth | 11.17 | Dieppe | 22.49 | | Senegal | 1.6 | Plymouth | 11.36 | Almirantes Eagle L. | 22.49 | | Callao | 5.8 | Avranche | 10.5 | Portsmouth | 22.57 | | Halifax | 4.14 | Enaowe | 11.44 | Ostend | 22.57 | | Marquesas | 9.16 to 11.46 | St Maloes | 12.16 | Nieport | 22.68 | | Quebec | 4.44 | Londonderry | 12.14 | Gravelines | 22.72 | | Cape La Hogue | 0.8 | Tonga Taboo | 12.16 | Alderney | 22.72 | | Gibraltar | 0.20 | Granville | 12.35 | Bergen | 22.72 | | Tahoga, Pan. Bay | 5.22 | Pudessa | 12.59 | False Bay | 22.72 | | Funchal | 1.8 | St Francisco | 13.10 | False Bay | 22.72 | | Porto Alto | 5.19 | Cork | 13.19 | Drostenheim | 22.72 | | Cape Bejaia | 0.53 | Bristol | 13.25 | Rotherhithe | 22.72 | | Chichester R. | 6.17 | Barfleur | 13.37 | Aberdeen | 22.72 | | P. of Wales Fort | 6.17 | Canterbury | 13.37 | North Caps | 22.72 | | Terceira | 1.49 | Venus Pt. Otag | 16.10 | Leith and Edinburgh | 22.72 | | New York | 4.57 | Mauritius | 16.10 | Amsterdam | 22.72 | | Cape Hanlopen, Virg. | 5.1 | Lizard | 16.10 | Rotterdam | 22.72 | | Cadiz | 0.25 | Noonts Sound | 14.1 | London Bridge | 22.72 | | Karakakoa Bay | 10.24 | Guernsey | 14.5 | Archangel | 22.72 | | Virgin Cape, Pat. | 4.32 | Palo Cordore | 14.9 | Bordeaux | 22.72 | | Valparaiso | 4.49 | Calcutta | 14.32 | Hamburg | 22.72 | | Cape Charles | 4.57 | Seychelles Alm. | 14.21 | Bremen | 22.72 | | Goree Island | 1.10 | Stromness | 15.10 | Antwerp | 22.72 | | York Fort | 6.9 | N. Zealand Q.C.S. | 15.19 | Scott Head | 22.72 | | Lisbon | 0.37 | Honfleur | 15.22 | Lynn | 22.72 | | Nantes, Rhône | 0.6 | Havre | 15.26 | Hague | 22.72 | | Tanna | 11.19 | Caen | 15.41 | Leotoffe | 22.72 | | | | | | London Bridge | 22.72 |

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**Note:** The table lists various locations around the world along with their corresponding longitude and high water times (H.W. Gr.T.). 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 604, or to 256 217: and the former disturbing force is to the whole of this as 2 to 604 at the point nearest the moon, and the second as 1 to 604 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 \( \frac{3}{2} \cdot \frac{P^3}{70} = \frac{3}{140} P^3 \); or, if \( z \) be the moon's zenith distance from any point of the surface, \( f = \frac{3}{70} P^3 \sin \cos z \).

**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 \( 2 \sin \cos 45^\circ = 1 \); 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 20 839 000 feet, the greatest height of a lunar tide in equilibrium would be 2-0166 feet, and that of a solar tide 8097; that is, suppos- ing the moon's horizontal parallax about 57', and her mass \( \gamma_0 \) of that of the earth.

**Theorem C. [G.]** 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 sphere 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 \( ex^2 \), 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 \( \cos \psi \), if \( \psi \) be the angular distance of the circle from its pole; and calling \( \phi \) the distance of this pole from the equatorial plane of the spheroid, the distance in question will be \( \cos \psi \sin \phi \), and \( x^2 + x'^2 = 2 \cos \psi \sin \phi \); and the difference \( x^2 - x'^2 \) is twice the actual sine of the arc \( \delta \) in the supposed circle, that is, twice the natural sine, reduced in the ratio of unit to the radius of this circle, which is \( \sin \psi \), reduced again to a direction perpendicular to the equatorial plane; whence \( x^2 - x'^2 = 2 \sin \psi \sin \phi \cos \phi \); and \( x^2 + x'^2 = 4 \sin \psi \sin \phi \cos \phi \). 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 \( \phi \); con- sequently 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 \( 4e \sin \psi \sin \phi \cos \phi \sin \psi \cos \phi \), re- duced first according to the distance and direction of each particle from the given gravitating particle; and we must compare the fluent with \( \frac{4}{3} \sigma \), the attraction of the whole sphere at the distance of the radius, or unity. But for the angle \( \delta \), the portion of the force acting in the common di- rection of \( \sin \psi \) is to the whole attraction at the same dis- tance as \( \sin \psi \) to 1, so that the attractive force of any point of the semicircle will be \( 4e \sin^3 \psi \sin \phi \cos \phi \sin \psi \cos \phi \), and its fluxion will be as \( \sin^3 \psi \cos \phi \), of which the fluent is \( \frac{4}{3} \psi - \frac{1}{2} \sin \phi \cos \phi \), or when \( \psi = 180^\circ, \frac{1}{2} \pi \), and \( \frac{1}{2} \pi \sin \psi \) will express the effect of the disturbing attraction of the semi- circles, of which \( \sin \psi \) is the radius, reduced to the direc- tion of the middle point, of which the distance is \( 2 \sin \frac{1}{2} \psi \); the reduction for this distance is as its square to 1; and for the direction, as the distance to \( \sin \psi \), together making the ratio of \( \frac{\sin \psi}{8 \sin^3 \frac{1}{2} \psi} \), and the ultimate fluxion of the force will be \( 2e \sin \psi \sin \phi \cos \phi \sin \psi \cos \phi \sin \psi \cos \phi \)

\[ d\psi = 2e \sin^3 \psi \cos \phi \sin \phi \cos \phi \]

\[ \cos \frac{1}{2} \psi \text{ and the fraction becomes } \frac{8 \sin^3 \psi \cos^3 \frac{1}{2} \psi}{8 \sin^3 \frac{1}{2} \psi} \cos \psi = \cos^3 \frac{1}{2} \psi \cos \psi = \cos^3 \frac{1}{2} \psi (\cos^3 \frac{1}{2} \psi - \sin^3 \frac{1}{2} \psi) = \cos^3 \frac{1}{2} \psi - \cos^3 \frac{1}{2} \psi + \cos^3 \frac{1}{2} \psi = 2 \cos^3 \frac{1}{2} \psi - \cos^3 \frac{1}{2} \psi. \]

Now, tak- ing the fluent from \( \psi = 0 \) to \( \psi = 180^\circ \), we have \( 2 \int \cos^3 \frac{1}{2} \psi \)

\[ \frac{d\psi}{2} \frac{1}{2} \psi = \frac{8}{5} \frac{4}{3}, \text{ and } \int \cos^3 \frac{1}{2} \psi \times 2d\psi = \frac{4}{3} \text{ the differ- } \] ence being \( \frac{3}{5} \cdot \frac{4}{3} = \frac{4}{5} \), whence the fluent of the force is found

\[ 2e \sin \cos \varphi \times \frac{4}{5} \times \frac{1}{n} \]

calling the density of the fluid \( \frac{1}{n} \)

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

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

\( \frac{3}{5} \) to \( \frac{5}{3} \); 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 \( \frac{3}{5} e \) must be to the relative force \( e \) as

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

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 \( f = 8-097 \) and 2-0166 respectively. But this is obviously far from the actual state of the problem.

Corollary 2. Supposing \( n = 5/4 \) (see Quarterly Journal of Science, April 1820), we have \( \frac{3}{27} + f = e \), 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 I. 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 point 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. Tides.

Sect. III.—Of the Effects of Resistance in Vibrating Motions, whether Simple or Compound.

Theorem G. If \( dw + Ads + Bds + Deds = 0 \), we have \( e^{Ds} (w + \frac{B}{D}s + \frac{AD - B}{DD}) = c \); \( h \) 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 \( \frac{x}{y} = \text{hly} - \text{lly} \), 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 fluents of different orders, or their coefficients. Such equations require in general to be compared with some multiple of the exponential quantity \( e^{nt} \), which affords fluents of successive orders, that have simple relations to each other, especially when \( dt \) is considered as constant. The multiples of \( \sin Ct \) 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 \( e^{nt} (w + ps + q) \) is

\[ e^{nt} (dw + pds + (nw + nps + ng) ds) = e^{nt} (dw + (p + ng) ds + nds + neds + neds); \]

and comparing with this \( e^{nt} (dw + Ads + Bds + Deds) \), we have \( n = D \), \( np = B \), and \( p = \frac{B}{n} = \frac{B}{D} \); and, lastly, \( p + ng = A \), \( q = \frac{A - p}{n} = \frac{AD - B}{DD} \); consequently the fluxion of \( e^{Ds} (w + \frac{B}{D}s + \frac{AD - B}{DD}) \) 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 \( \frac{dds}{dt^2} + Bs - D \frac{ds^2}{dt^2} = 0 \).

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 \( \frac{3g}{l} \) 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 \( ds \frac{dds}{dt^2} + Bds - D \frac{ds^2}{dt^2} ds = 0 = \frac{1}{2} dw + Bds - Dws \), and \( dw + 2Bds - 2Dws = 0 \); which, compared with the theorem, gives us \( 0 \) for \( A \), \( 2B \) for \( B \), and \( -2D \) for \( D \); and the solution becomes

\[ e^{-2Ds} \left( w - \frac{B}{D} - \frac{B}{2DD} \right) = c, \quad \text{or} \quad w = \frac{B}{D} + \frac{B}{2DD} + ce^{2Ds}; \]

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

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

\[ = -\beta e^{2D(s-\lambda)}, \quad \text{will become} \quad -\beta e^{-2Ds}, \quad \text{and} \quad w = \frac{B}{D} + \frac{B}{2DD} - \beta e^{-2Ds} = \frac{B}{2DD} (1 - \gamma e^{-2Ds} + 2Ds). \]

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

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

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

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 \( Bds = Dws \), and \( Bs = Dw \), which is the obvious expression of the equality of the re- sistance to the propelling force. Putting the greatest value of \( w = x \), and the corresponding value of \( s = z \), we have

\[ \frac{B}{D} + \frac{B}{2DD} + ce^{2D} = \frac{B}{D} + \frac{B}{2DD} \]

since \( B = Dz \), and \( \frac{B}{2DD} = -ce^{2D} \); hence \( \frac{1}{\gamma} = e^{2D}(\lambda - \lambda) \),

and \( \gamma = e^{2D}(\lambda - \lambda) \); consequently \( hl = 2D(\lambda - \lambda) = hl \)

\[ (1 + 2D\lambda), \text{ and } 2Dz = 2D\lambda - hl (1 + 2D\lambda), \text{ and } z = \frac{1}{2D} \left( 2D^2\lambda^2 - \frac{8}{3} D^3\lambda^3 + \ldots \right) = D\lambda^2 - \frac{4}{3} D^3\lambda^3 + \ldots \]

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

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

The proof of this well-known formula is the most readily obtained by means of a series with indeterminate co-efficients, such as \( z = Az + Bz^2 + \ldots \), 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 \( Be \), the equation \( 2\lambda = \gamma s - \frac{2}{3} D\gamma s^2 + \frac{1}{3} D^2\gamma s^3 - \ldots \); and, by reversing this series, we have \( \epsilon = \frac{2\lambda}{\gamma} + \frac{8}{3} D\lambda^2 + \frac{40}{9\gamma} D^2\lambda^3 \ldots \), or \( \epsilon = 2\lambda - \frac{4}{3} D\lambda^2 + \ldots \);

the difference of the arcs of descent and ascent being \( \frac{4}{3} D\lambda^2 \), and the difference of two successive vibrations \( \frac{8}{3} D\lambda^3 \), when the resistance is very small; this difference being also \( \frac{8}{3} \); 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 \( s = z = r \), and \( x = w = z \), then \( s = r + z, z = \lambda - s - r \),

\[ w = \frac{B}{2DD} (1 - \gamma e^{-2D(\lambda - \lambda)} + 2D[s + r]), \text{ and } z \]

being \( \frac{B}{D} z = -\frac{B}{2DD} - \frac{B}{D} r + \frac{B}{2DD} e^{2D(\lambda - \lambda)} e^{2Dr} \),

but we have seen that \( e^{2D}(\lambda - \lambda) = \frac{1}{\gamma} \), and \( z \) becomes

\[ \frac{B}{2DD} + \frac{B}{D} r - \frac{B}{2DD} e^{2Dr} = \frac{B}{D} - \frac{B}{2DD} (2Dr + 2D^2r^2 + \ldots) \]

\[ + \frac{4}{3} D^3r^3 + \frac{8}{12} D^4r^4 + \ldots = \frac{B}{D} r - \frac{B}{D} r - Br^2 - \frac{2}{3} BD^2r^3 - \frac{1}{3} BD^3r^4 - \ldots, \]

and

\[ \frac{z}{B} = r^2 + \frac{2}{3} Dr^3 + \frac{1}{3} D^2r^4 + \ldots \]

In order to reverse this series, we must put \( \frac{z}{B} = y^2 \), and \( r = \lambda y + ey^2 + cy^3 + \ldots \); and by substituting the powers of this series for those of \( r \) in the value of \( \frac{z}{B} = y^2 = r^2 + r^3 + \ldots \),

we find \( \lambda = 1, n = -\frac{1}{2} p, c = \frac{5}{8} r^2 - \frac{1}{2} q \ldots \); and \( r = y \)

\[ -\frac{1}{3} Dy^2 + \frac{1}{9} D^2y^3 - \ldots \]

Hence \( dr = dy - \frac{2}{3} Dydy + \frac{1}{3} D^2y^2dy - \ldots \); and this fluxion, divided by the velocity \( v = \sqrt{(x - z)} \), will be the fluxion of the time; or, since

\[ \frac{dz}{B} = 2ydy \text{ and } dy = \frac{dz}{2By} = \frac{dz}{2\sqrt{B}v}, \]

\[ dt = \frac{1}{B} \cdot \frac{dz}{2\sqrt{(x-z)}} - \frac{D}{3B} \cdot \frac{dz}{\sqrt{(x-z)}} + \frac{D^2}{6B\sqrt{B}} \cdot \frac{dz}{\sqrt{(x-z)}} \ldots \]

and the fluent becomes \( t = \frac{1}{2\sqrt{B}} \arctan \frac{2z}{x} - \frac{2D}{2B} \sqrt{(x-z)} + \frac{D^2}{6B\sqrt{B}} \left( \frac{1}{2} \arctan \frac{2z}{x} - \frac{\sqrt{(x-z)}}{x} \right) \ldots \); the value of which, taken from \( z = 0 \) to \( z = x \), is \( \frac{\pi}{2\sqrt{B}} - \frac{2D}{3B} \sqrt{x} + \frac{D^2}{6B\sqrt{B}} (\frac{1}{2}\pi) \ldots \)

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 \( \frac{\pi}{2\sqrt{B}} + \frac{2D}{3B} \sqrt{x} + \frac{D^2}{12B\sqrt{B}} \pi + \ldots \),

the sum of both being \( \frac{\pi}{\sqrt{B}} + \frac{D^2\pi}{6B\sqrt{B}} + \ldots \), which is the time of a complete vibration, and the difference \( \frac{4D}{3B} \sqrt{x} + \ldots \)

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 \( \frac{\pi}{\sqrt{B}} \).

Theorem H. If \( \frac{dz}{dt} + A \frac{dz}{dt} + Bs = 0 \), \( dt \) being con- stant, we have \( e^{mt} (ds + adt) = c \); \( m \) being \( \frac{1}{2} A = \sqrt{\left( \frac{1}{4} A^2 - B \right)} \), and \( a = \frac{1}{2} A = \sqrt{\left( \frac{1}{4} A^2 - B \right)} \).

**Demonstration.** The fluxion of \( e^{mt} (ds + adt) \) is \( e^{mt} \)

\[ \left( d^2s + adst + \left( mds + amdt \right) dt \right) = e^{mt} \left( d^2s + \left( a + m \right) dsdt + amdt^2 \right); \]

and, comparing this fluxion with the proposed equation, we have, for the co-efficients, \( a + m = A \),

and \( am = B \); whence \( \frac{B}{m} + m = A \), \( m^2 - Am = -B \),

\( m = \frac{1}{2} A = \sqrt{\left( \frac{1}{4} A^2 - B \right)} \), and \( a = \frac{1}{2} A = \sqrt{\left( \frac{1}{4} A^2 - B \right)} \).

**Example.** Let the equation proposed be that of a cycloidal pendulum, vibrating with a resistance proportional to the velocity; that is,

\[ \frac{d^2s}{dt^2} + A \frac{ds}{dt} + Bs = 0. \]

**Scholium 1.** The resistance is here adequately expressed, in all cases, by the term \( A \frac{ds}{dt} \) so that the equation is permanently applicable to the successive vibrations. Thus, in the second descent, on the negative side of the vertical line, \( Bs \) being negative, and \( s \) becoming nearer to 0, the fluxion \( ds \) is positive, and \( A \frac{ds}{dt} \) is of a contrary character to \( Bs \), as it ought to be.

**Solution.** Since \( m = \frac{1}{2} A = \sqrt{\left( \frac{1}{4} A^2 - B \right)} \), and \( a = \frac{1}{2} A = \sqrt{\left( \frac{1}{4} A^2 - B \right)} \), it is obvious that the two radical quantities will be either possible or imaginary, according as \( \frac{1}{4} A^2 \) is greater or less than \( B \).

**Case i.** If \( A^2 \) is greater than \( 4B \), the resistance being very considerable, the solution becomes

\[ e^{\frac{1}{2} At} \pm \sqrt{\left( \frac{1}{4} A^2 - B \right)} \left( \frac{ds}{dt} + \left[ \frac{1}{2} A \mp \sqrt{\left( \frac{1}{4} A^2 - B \right)} \right] s \right) = c; \]

and the velocity \( v = -\frac{ds}{dt} = \left( \frac{1}{2} A \mp \sqrt{\left( \frac{1}{4} A^2 - B \right)} \right) s - e^{\frac{1}{2} At} \mp \sqrt{\left( \frac{1}{4} A^2 - B \right)} t \);

and if the velocity be supposed to vanish when \( s = \lambda \), and \( t = 0 \), we have \( 0 = \frac{1}{2} A \lambda - \sqrt{\left( \frac{1}{4} A^2 - B \right)} \lambda \), \( c = \frac{1}{2} A \lambda + \sqrt{\left( \frac{1}{4} A^2 - B \right)} \lambda \).

**Corollary 1.** Hence it appears that such a pendulum would require an infinite time to descend to the lowest point, since the velocity cannot have a finite value when \( s \) vanishes, the exponential quantity never becoming negative.

**Scholium 2.** The co-efficient \( B \) may also be written for an actual pendulum, as measured in English feet, \( \frac{32}{T} \) or \( \frac{2g}{T} \), if we call \( g \) the descent of a falling body in the first second, which is, however, denoted in the works of some authors by \( \frac{1}{2} g \), or even by \( \frac{1}{4} g \). If we make \( \frac{d^2s}{dt^2} + \frac{32}{T} s = 0 \);

when \( s = l \), the force becomes such that \( -d^2s = 32ds \),

and \( -\frac{ds}{dt} = 32 \), which is the true velocity generated by such a force in a second of time. Supposing \( h \) to be the velocity with which the resistance would become equal to the weight, we must have for \( A \frac{ds}{dt} \frac{32}{k} \frac{ds}{dt} \) in order that the force represented by \( A \) may become equal to that of gravity, and \( A = \frac{2g}{k} \); and if \( h \) be the height from which a body must fall to gain the velocity \( h \), since \( h = \frac{k^2}{4g} \), \( A = \frac{4gg}{hk} = \frac{g}{h} \).

Hence it follows, that when \( A^2 = 4B \), which is the time of the possibility of alternate vibrations, \( \frac{g}{h} = \frac{8g}{T} \), and \( h = \frac{1}{8} T \),

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

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

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

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

or,

since \( \sqrt{-1} = \frac{1}{\sqrt{-1}} \),

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

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, \( \cos. Ct \left( \frac{ds}{dt} + \frac{1}{2} As \right) + \sin. Ct Cs = ce^{-\frac{1}{2} At} \);

whence

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

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 \( \frac{1}{2} As = v = \frac{1}{2} As - s_0 \); consequently \( c = 0 \). The equation will then become

\[ \frac{ds}{s} + \left( \frac{C \sin. Ct}{\cos. Ct} + \frac{1}{2} A \right) dt = 0; \]

whence

\[ \ln s - \ln \cos. Ct = c' - \frac{1}{2} At = \ln \frac{s}{\cos. Ct} \quad \text{and} \quad \frac{s}{\cos. Ct} = e^{c' - \frac{1}{2} At}; \]

or \( s = \cos. Ct \cdot e^{c' - \frac{1}{2} At} \); and, when \( t = 0 \), \( s = e^{c'} \); consequently \( \frac{s}{\cos. Ct} = \cos. Ct \cdot e^{-\frac{1}{2} At} = \cos. Ct \left( 1 - \frac{1}{2} At + \frac{1}{8} A^2t^2 - \frac{1}{48} A^3t^3 + \cdots \right) \).

But since \( v = -\frac{ds}{dt} = \left( \frac{C \sin. Ct}{\cos. Ct} \cdot dt + \frac{1}{2} Adt \right) \), 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 \( -\frac{1}{2} At = \frac{AA}{4CC} \), and \( e^{-\frac{1}{2} At} = 1 + \frac{AA}{4CC} \), very nearly; so that, calling the primitive extent of the arc of vibration \( s = \lambda \), we have \( \frac{\lambda}{s} = \cos. Ct \left( 1 + \frac{AA}{4CC} \right) \); \( \cos. Ct \)

\[ \frac{2k}{VOL. XXL} \] Tides.

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

\( \lambda = 1 + \frac{AA}{8CC} \), and \( \frac{\varepsilon}{\lambda} = \frac{8CC}{8CC + AA} = \frac{8B - 2AA}{8B - AA} = 1 - \frac{AA}{8B - AA} \), 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 \( e \) 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. \( Ct = \frac{A}{\sqrt{B}} = \cos. Ct = \frac{\varepsilon}{\lambda} \) 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 \( 1 - \frac{A}{C} \varepsilon \), and \( \frac{A}{C} \sigma \),

or \( \frac{A\sigma}{\sqrt{B}} \) 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 \( A = \frac{1}{1800} \times 4054651 = 00022526 \), and \( A^2 = 00000005075 \); and since \( B = \frac{32}{\lambda} = 981 \), \( C = \sqrt{(B - \frac{1}{4}A^2)} = \sqrt{B} \sqrt{1 - \frac{AA}{4B}} = \sqrt{B} \left(1 - \frac{AA}{8H}\right) \); the fraction being only \( = 0000000065 \); 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 apparent, tempora tam ascensuum quam descensuum inter se esse aquilia, tamen determinari non potest, quantum sit temporis sine descensuum sine ascensuum: neque enim tempora descensuum et ascensuum inter se possint comparari.* Equatio enim rationem inter s et u difficilius ita est complicata, ut ex ea elementum temporis \( \frac{ds}{u} \), per unicum variabilem non posset 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 \( \frac{ds}{dt} = e^{mt} (m \cos. Ct - C \sin. Ct) \), and \( \frac{d^2s}{dt^2} = e^{mt} (m^2 \cos. Ct - Cm \sin. Ct - Cm \sin. Ct - Cm \cos. Ct) \); whence \( \frac{d^2s}{dt^2} + A \frac{ds}{dt} + Bs = e^{mt} (m^2 \cos. Ct - 2Cm \sin. Ct - C^2 \cos. Ct + Am \cos. Ct - AC \sin. Ct + B \cos. Ct) = 0 \), and \( (m^2 - C^2 + Am + B) \cos. Ct - (2Cm + AC) \sin. Ct = 0 \): 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 second vibration, 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 \( \frac{A}{\sqrt{B}} \sigma \lambda \), 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 $\frac{1}{2}x - \frac{1}{3}\sin x\cos x$, and that of the space by $\frac{1}{2}x^2 - \frac{1}{3}\sin^2 x$, which, at the end of a vibration, becomes $\frac{1}{2}x^2 - \frac{1}{3}\sin^2 x$, which is, since the distance of the point of greatest velocity is here $s = D\lambda^2$, $\frac{1}{2}D\lambda^2 = 2467 D\lambda^2$, while the more accurate mode of computation has shown that the true diminution of the space is $2667 D\lambda^2$. (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 a synchronous vibration, as hereafter in Theorem K, Schol. 1.

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

**Demonstration.** The value of $s$ here assigned gives us

$$\frac{d^2s}{dt^2} = \sqrt{B}\cos \sqrt{Bt} + \frac{MF}{FF-B}\cos Ft,$$

and

$$\frac{d^2s}{dt^2} = -B\sin \sqrt{Bt} - \frac{MFF}{FF-B}\sin Ft;$$

so that

$$\frac{d^2s}{dt^2} + Bs = B\sin \sqrt{Bt} + \frac{MB}{FF-B}\sin Ft - \frac{MFF}{FF-B}\sin Ft = -M\sin Ft.$$

**Corollary 1.** If, in order to generalize this solution, we make $s = a\sin \sqrt{Bt} + b\cos \sqrt{Bt} + c\sin Ft + d\cos Ft$, 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 $\sqrt{Bt}$, 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 $M'\sin Ft$, $M''\sin F't$, ..., will add to the solution the quantities $\frac{M}{FF-B}\sin Ft$,

$$\frac{M'}{FF'-B}\sin F't,$$ 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 $\frac{1}{T}$, 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 $s = \frac{M}{FF-B}(\sin Ft - \cos \sqrt{Bt})$; 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 $\sin^2 x = .8484 \sin x - .01696 \sin 3x - .00424 \sin 5x - .00013 \sin 7x - .00029 \sin 9x - .00013 \sin 11x - ...$. 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... Tides.

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

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 $(B - G^2) = \beta AG + M = 0$, and $(B - G^2) \beta + \alpha AG = 0$; whence $\beta = \frac{AG}{GG - B}$, and consequently $\alpha = \frac{(GG - B)M}{(GG - B)^2 + AAGG}$,

and $\beta = \frac{AGM}{(GG - B)^2 + AAGG}$. And since, in general, if $b = ta. b$, $\sin. x + b \cos. x = \sqrt{(1 + b^2)} \sin. (x + b)$; $\sin. (x + n)$ being $\sin. x \cos. b + \sin. n \cos. b = \cos. b (\sin. x + ta. b \cos. x)$, and therefore $\sin. x + ta. b \cos. x = \frac{\sin. (x + b)}{\cos. b} = \sin. (x + b) \sec. b = \sin. (x + b) \sqrt{(1 + b^2)}$;

it follows that $\alpha \sin. Gt + \beta \cos. Gt = \alpha (\sin. Gt + \arctan \frac{\beta}{\alpha}) \sqrt{(1 + \beta^2)}$; and $\alpha \sqrt{(1 + \beta^2)} = \sqrt{(\alpha^2 + \beta^2)} = \frac{M}{\sqrt{(GG - B)^2 + AAGG}}$.

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

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

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

$$\left(Gt + \arctan \frac{\beta'}{\alpha'}\right) = \sqrt{(\alpha'^2 + \beta'^2)} \sin.$$

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

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 $AG^2$, 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 tan. $\frac{AG}{B - GG}$ 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 co-efficient, 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 $M = \frac{2g}{l} \mu = 32 \frac{\mu}{l}$, and $\mu = \frac{ML}{2g} = \frac{1}{32} ML$.

Corollary. In order to obtain a more general solution of the problem, we may combine the periodical motion 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 $s = \alpha \sin. Gt + \beta \cos. Gt + \alpha' \sin. Ft + \beta' \cos. Ft + ... = \sqrt{(\alpha^2 + \beta^2)} \sin.$

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

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

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 \( \cos c \cos b = \frac{1}{2} \cos (a + b) + \frac{1}{2} \cos (a - b) \), 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 \( \cos d - \cos c = 2 \sin \frac{c + d}{2} \cos \frac{c - d}{2} \); also \( \sin a + \sin b = 2 \sin \frac{a + b}{2} \cos \frac{a - b}{2} \); and \( \sin a - \sin b = 2 \sin \frac{a - b}{2} \cos \frac{a + b}{2} \).

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

\[ \frac{\frac{1}{2} R}{\sqrt{[(F - G)^2 - B]^2 + A^2 (F - G)^2}} \quad \text{and} \quad \beta = \frac{\frac{1}{2} R}{\sqrt{[(F + G)^2 - B]^2 + A^2 (F + G)^2}} \]

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

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

Sect. 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 tides 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 \( \sin \cos Alt \sin Az \), if the canal be in an easterly and westerly direction; or if it deviate from that direction in a given angle, 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 as 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, \( \sin Az = \frac{\cos Decl \cdot \sin Hor}{\cos Alt} \), and the expression, \( \sin \cos Alt \cdot \sin Az \), becomes in this case \( \sin Alt \cdot \cos Decl \cdot \sin Hor \cdot \cos Decl \cdot \sin Hor \); but \( \sin Alt = \sin (Lat) \cdot \sin Decl + \cos (Lat) \cdot \cos Decl \cdot \sin Hor \), and calling \( \sin (Lat) \) for the given canal \( l \), and \( \cos (Lat) \), the force becomes \( l \cdot \sin \cos Decl \cdot \sin Hor \cdot \cos Decl \cdot \sin Hor \cdot \cos Decl \cdot \sin Hor \cdot \cos Decl \cdot \sin Hor \); now, \( \sin Decl = \cos Obl \cdot Ecl \cdot sin Lat + \sin Obl \cdot Ecl \cdot cos Lat \cdot sin Long \); and since \( \cos \phi = 1 - \frac{1}{2} \sin^2 \phi + \frac{3}{8} \sin^4 \phi - \frac{5}{16} \sin^6 \phi + ... \), 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 \( \varphi \), the horary angles \( t \) and \( t' \), and the sine of the obliquity of the ecliptic \( \alpha \), we shall have \( \sin Decl = \alpha \sin \Theta \), or \( \alpha \sin \varphi \); and \( \cos Decl = 1 - \frac{1}{2} \alpha^2 \sin^2 \Theta + \frac{3}{8} \alpha^4 \sin^4 \Theta + ... \), and \( \sin \cos Decl = \alpha \sin \Theta - \frac{1}{2} \alpha^3 \sin^3 \Theta + \frac{3}{8} \alpha^5 \sin^5 \Theta + ... \); also \( \cos^2 Decl = 1 - \alpha^2 \sin^2 \Theta \); whence the sun's force becomes \( l \cdot \sin t (\alpha \sin \Theta - \frac{1}{2} \alpha^3 \sin^3 \Theta + \frac{3}{8} \alpha^5 \sin^5 \Theta + ...) + \frac{1}{2} l' \sin 2t (1 - \alpha^2 \sin^2 \Theta) = l \cdot \sin t (\alpha \sin \Theta - \frac{1}{2} \alpha^3 \sin^3 \Theta + \frac{3}{8} \alpha^5 \sin^5 \Theta + ...) + \frac{1}{2} l' \sin 2t (1 - \alpha^2 \sin^2 \Theta) = l \cdot \sin t (\alpha \sin \Theta - \frac{1}{2} \alpha^3 \sin^3 \Theta + \frac{3}{8} \alpha^5 \sin^5 \Theta + ...) + \frac{1}{2} l' \sin 2t (1 - \alpha^2 \sin^2 \Theta) \).

\( \alpha^2 = \frac{1}{8} \alpha^2 - \frac{15}{128} \alpha^5 = 0.078 \), and \( \alpha^4 = \frac{3}{128} \alpha^5 = 0.0002 \), and \( \alpha^6 = 1.585 \). 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 \( S \left( l \cdot \alpha^2 \left[ \frac{1}{2} \cos (t - \Theta) - \frac{1}{2} \cos (t + \Theta) \right] + l \cdot \alpha^4 \left[ \frac{1}{2} \cos (t - 3\Theta) - \frac{1}{2} \cos (t + 3\Theta) \right] + l \cdot \alpha^6 \left[ \frac{1}{2} \cos (t - 5\Theta) - \frac{1}{2} \cos (t + 5\Theta) \right] \right) + \frac{1}{2} (1 - \alpha^2 \sin^2 \Theta) \sin 2t + \frac{1}{2} \alpha^2 \left[ \frac{1}{2} \sin 2(t + \Theta) + \frac{1}{2} \sin 2(t - \Theta) \right] \); and that of the moon may be expressed in the same manner, by substituting \( M, t' \), and \( \varphi \), 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 $\frac{1}{2} v'$ sin $2t$, and $\frac{1}{2} v' \sin 2t'$, 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 greatest, it will be $D (M' - S)^2$, and when least, it will be $D (M' + S)^2$, 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, $A' = A + 288 DH'$, 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

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

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 $B$. 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 $\frac{1}{2} \cos (t - \Theta) + \frac{1}{2} \beta \sin (t - \Theta)$, 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 $\frac{1}{2} \alpha [\cos (t - \alpha) - \cos (t + \alpha)] + \frac{1}{2} \beta [\sin (t - \Theta) - \sin (t + \Theta)] = \alpha \sin t \sin \Theta + \beta \cos t \sin \Theta = \sin \Theta (\alpha \sin t + \beta \cos t)$; which is the same as if we considered the effect of the force $\sin t$ 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 $L \sin Gt$, $Deel. \sin Hor.$ $< + \frac{1}{2} L \cos^2 Decl. \sin. cos. Hor.$ $< + \frac{1}{2} L \sin 2 Decl. \sin Hor.$ $< + \frac{1}{2} L \cos^2 Decl. \sin 2 Hor.$ $<$, 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 those 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 \( \epsilon \) will be found from the variation, or the moon's age in space.

Now, in Burckhardt's Tables, p. 87, we find the variation for the midnight ending 1823, by adding the constant quantity \( 9^\circ \) to the epoch for 1824, and (11° 14' 44" 44") + \( 9^\circ = 11^\circ 23' 44'' 44'' \), or — (6° 15' 16"), according to the time of Paris. The movement for 12 hours is 6° 5' 43"; consequently at noon, or 1824 Jan. 1, 0h. astronomical time at Paris, the variation is — (9 33"), corresponding to the movement of 18m. 49s. in mean time, and the mean conjunction will take place at 18m. 49s. Parisian time, which may be more commodiously 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° 11' 26.5" = 12° 19", we shall have \( \epsilon = 360^\circ - 12^\circ 19" = 347^\circ 81" \) when \( t = 360^\circ \), or \( \epsilon = \frac{347^\circ 91}{360} = 96614t \); and the moon's horary angle, considered in relation to the circumference as unity, will always be 96614t, if \( t \) be 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° 25' 23.1") = 77666; his longitude for any other time will therefore be 77666 + 002738t = \( \Theta \), and that of the moon \( \varphi = 77666 + 03386t \).

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° 29' 25" 3.3") + 2° = 151° 25' 23.3"; the movement for 12h. 18m. 49s. is (6° 31' 57") + (9° 47' 49") + 27" = 6° 42' 12", which gives 158° 7' 35" for the mean anomaly at noon in the island of Guernsey. The daily movement being 13° 3' 9" = 13° 065", the mean anomaly will always be 158° 127" + 13° 065t, reckoning \( t \) from the supposed epoch or day. The principal part of the central equation will then be, according to Burckhardt, 226924° sin. \( An \), or (6° 18' 2") sin. (158° + 13° 065t), and its sine will be very nearly 11 sin. (13° 065t + 158° 127"), which will represent the principal inequality of the longitude and of the variation, so that the variation, instead of 12° 19", will become 12° 19" + 6° 3" sin. (13° 065t + 158° 127"), and this subtracted from 360°, leaves 347° 81" — 6° 3" sin. (13° 065t + 158° 127"), the sine of which is nearly sin. 347° 81" — cos. 347° 81" + 11 sin. (13° 065t + 158° 127").

The equatorial parallax is nearly 57" + 187" cos. \( An \), or 57" + 3° 1' cos. (13° 065t + 158° 127"); and the disturbing force, which varies as the cube of the parallax, or of 57" [1 + .0544 cos. (13° 065t + 158° 127")], may be expressed, with sufficient accuracy, by 1 + .1632 cos. (13° 065t + 158° 127").

The supplement of the node for 1824 is (2° 10' 56") + 2° = 70° 58", to which we must add (3° 10' 6") \( t \) for the time elapsed; and the longitude \( \varphi \) will be 279° 35' 23.1" + (13° 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 \( \frac{1}{2} \) and \( \frac{1}{10} \), supposing \( D \) to be inconceivable. We then find, from the expression \( \sqrt{(a^2 + b^2)} M = \sqrt{(a^2 + b^2)} B_{\mu} \) (Theorem K, Schol. 2) = \( \frac{B(SM)}{\sqrt{(GG - B)^2 + AAGG}} \) for the solar tide, \( G \) being 1, if

\[ B = \begin{cases} \frac{1}{2}, & 93442, \\ \frac{1}{10}, & 980, \\ \frac{1}{3}, & 832, \\ \frac{1}{4}, & 742, \\ \frac{1}{5}, & 1122, \\ \frac{1}{6}, & 913, \end{cases} \]

1, or 4;

\[ A = \begin{cases} \frac{1}{10}, & 93442, \\ \frac{1}{10}, & 980, \\ \frac{1}{3}, & 832, \\ \frac{1}{4}, & 742, \\ \frac{1}{5}, & 1122, \\ \frac{1}{6}, & 913, \end{cases} \]

1, or 4;

and for the lunar, \( G \) being 96614, and

\[ A = \begin{cases} \frac{1}{10}, & 93442, \\ \frac{1}{10}, & 980, \\ \frac{1}{3}, & 832, \\ \frac{1}{4}, & 742, \\ \frac{1}{5}, & 1122, \\ \frac{1}{6}, & 913, \end{cases} \]

1, or 4;

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 the equation

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

whence we find that \( B \) must be either .9380 or .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 = .93442, \( A \) cannot be greater than .6328, and \( B \) would then be .78540; and if \( A = 0 \), the values of \( B \) would be .9617 or .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 \( B = 9 \) and \( A \) still = \( \frac{1}{10} \), we should have 7.071 and 9.756 for their magnitude. Now if \( B = .6328 \), the tangents of the angular measures of the displacement,

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

become \( \frac{1}{96614} \) and \( \frac{1}{30160} \)

respectively, giving us 69° 50' and 72° 40' for the angles themselves; and if \( B = 9 \), these angles become 45° and 70° 24' respectively; the difference in the former case 2° 50', and in the latter 25° 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-