and and will fall during an equal time; the whole rise and Waves fall will usually be less than in rivers; a very slight current will be directed towards the head of the bay during the rise, and from it during the fall of the water. The variations of spring and neap tides, and the relaeftion of the time of high water to the time of the moon's passage over the meridian, follow the same general laws as in rivers. (9.) In long and narrow seas (for instance the English Channel) the tide in mid-channel follows the same laws as at a station near the mouth of a river, rising and fall**ing in equal times, and running in a direction which may be considered analogous to the direction up a river, for three hours before and three hours after high water; and in the opposite direction, for three hours before and three hours after low water. But near the sides of the channel, and especially near the mouths of bays or estuaries branching from the channel, the change of tide follows a of very peculiar law. The water is never stationary, as in river-tides, when changing from flow to ebb, but the the direction of the current changes in 12 hours 20 minutes through all the points of the compass. As a general rule, supposing the observer's face turned in the direction which is analogous to the direction up a river; near the shores on his left hand the course of the tidecurrent revolves in the same direction as the hands of a watch, and near the shores on the right hand it revolves in the opposite direction. Near the headlands which separate different bays, there is usually, at certain times of the tide, a very rapid current, called a race. (10.) The elevations and depressions of tides in the open seas are much smaller than in contracted seas or rivers, sometimes not exceeding one or two feet; the stream of the tide is generally insensible. 2 (11.) In seas of small extent (as the Mediterranean) the Tide is nearly insensible. (12.) In some circumstances, phænomena which are scarcely perceptible in ordinary localities become paramount. Thus, in some positions near Behring's Straits, the difference of morning and evening tides, which is scarcely sensible in England, becomes so great that, in ertain parts of the lunation, there appears to be only e tide in the day. Other phænomena peculiar to these localities, but less obvious to ordinary observation, will be noticed hereafter. the moon's age. (13.) The phænomena which we have described must necessarily, for the most part, have been remarked by all nations dwelling on the borders of the ocean. Thus Casar, in his account of the invasion of Britain, (De Bello Gallico, lib. iv.) alludes to the nature of spring tides as perfectly well understood in connection with Some of the peculiarities of river tides, however, were not published in scientific works till the beginning of the last century; and some of the properties of the tides in the English and other channels were not known till the end of that century. Upon the whole, the statement above may be supposed to represent pretty well all that was known of the Tides about the year 1800: and it will serve to point out to the reader the leading facts, to whose explanation a Theory of the Tides ought to be directed. In the present century, the elaborate discussions of immense collections of accurate tide-observations by M. Laplace, Sir John W. Lubbock, and Professor Whewell, have brought to light and reduced to law many irregularities which were before that time unknown. We prefer, however, delaying the particular mention of these until we have discussed the various forms in which theory Tides and has been put for the purpose of explaining the grand Waves. facts of the Tides. SECTION II.-EQUILIBRIUM-THEORY OF TIDES. theories of and the (14.) Before entering upon either of the theories Inade. explaining the Tides, we must allude to their inade- quacy of all quacy, perhaps not to the explanation of the facts the Tides, already observed, but certainly to the prediction of new ones. This inadequacy does not appear to arise cause of it. from any defect in the principles upon which the theory is based, (although perhaps our ignorance of the laws of friction among the particles of water, and between water and the sides of the channels which contain it, may be considered a failure of this kind,) but from the extreme difficulty of investigating mathematically the motions of fluids under all the various circumstances in which the waters of the sea and of rivers are found. For the problem of the Tides, it is evident, is essentially one of the motion of fluids. Yet so difficult are the investigations of motion that, till the time of Laplace, no good attempt was made to determine, by theory, the laws of the Tides, except on the supposition that the water was at rest. Since that time theories of motion have been applied; and it is hoped that in the present Treatise it will be found that something has been added to the preceding investigations of motion, possessing in some degree a practical character. Yet the theory, even in this state, reaches very few cases. Indeed, throughout the whole of this subject, the selection of the proper theoretical ground of explanation is a matter of judgment. In some cases we may conceive that we are justified in using the Equilibriumtheory; in others the Wave-theory will apply, completely or partially; in a few cases, the results of observation in one locality will be considered as a fundamental set of experiments, upon which the explanation of the phænomena in other localities will be grounded without further reduction to theory; and as a last resource, in almost every case, we shall be driven to the same arbitrary suppositions which Laplace introduced. Nevertheless, we conceive that our mathematical theory, pursued into some degree of detail, will be far from useless. In the instances which it does not master completely, it will show that there are ample grounds for the arbitrary alterations of constants introduced by Laplace in his suppositions, to which we have more than once alluded. It will show that we are precluded from further advance, partly by our almost necessary ignorance of the forms of the bottom in deep seas, and partly by the imperfection of our mathematics. It will leave no doubt whatever that the first principles of our explanation are correct. Begging the reader to receive the first part of this paragraph as an apology on the part of mathematicians for applying to the motion of Tides a theory so evidently inadequate as the Equilibrium-theory, we shall now proceed to give that theory, nearly in the terms of its proposers. equili (15.) The popular explanation of the Equilibrium- Popular theory is very simple. If we conceive the earth to be explanacovered wholly or in a great degree with water, and tion of consider that the attraction of the moon upon different briumparticles (according to the law of gravitation) is in- theory. versely as the square of their distance, and is therefore greatest for those particles which are nearest to it; then it will be obvious that the moon attracts the water Tides and Newton's tion of the sea. of motion. 386046 Wan Sect. Equili Newt vatio the v by th on that side which is next to her, more than she attracts 90° distant from those vertically under the sun, is Tides (16.) The theory of Newton is rather a collection of first theory hints for a theory than any thing else. In the Princiof the mo- pia, lib. I. prop. 66, cor. 19, he has (by a remarkable deduction from the Lunar Theory) considered the motion of water in a canal passing round the earth in or near to the earth's equator, and has arrived at the singular conclusion that the water would be lowest in that part which is most nearly under the body (the sun or moon) whose attraction causes the motion of the water. This conclusion we shall find to be entirely supported by more complete investigations. In lib. III. prop. 24, he has modified this conclusion, and His modi- seems to suppose that in free seas the high water ought fied theory to follow the moon's transit over the meridian (conceiving, for the moment, the moon's attraction to be the sole exciting cause of the Tides) in three hours, or at least in less than six hours. To this he appears to have been led by erroneous reasoning of the same kind as that which, in lib. I. prop. 66, cor. 20, has introduced an incorrect inference as to the Solar Nutation of the Earth's axis. We shall find hereafter that the introduction of friction into our theories of the motion of water will lead to a conclusion somewhat similar. The only part in which he uses numerical calculation is in lib. III. prop. 36, and 37, the subjects of which are, "Invenire vim Solis ad Mare movendum," "Invenire vim Lunæ ad Mare movendum." The following Newton's is his method of computation (the demonstration of the calculation different parts of which we defer till we treat of the more complete theory of Bernoulli). First he refers to the Lunar Theory for a calculation of the force which the sun exerts to draw the moon, when in quadratures, towards the earth, and he finds it to be part of gravity at the earth's surface. Then he remarks that the similar force upon the water at the earth's surface, in the position distant 90° of terrestrial arc from the point to which the sun is vertical, is less than the force upon the moon, in the proportion in which the water's distance from the centre of the earth is less than the moon's distance from the centre of the earth, or in the proportion of 1 : 60-5: and therefore the force which depresses the water, at the points of the force of the Sun. 289 of th (17.) In order to ascertain the effect which the No moon's action would produce, it is necessary to know ca the mass of the moon. For this there were in Newton's time no direct means: and he was, therefore, obliged to refer to the phænomena of the Tides themselves, as for observed in places where, from local causes, the rise of M the tide is very considerable. He quotes the observations of Sturmy on the tides in the Severn, at the mouth of the Avon, which give 45 feet for equinoctial spring tides, 25 feet for equinoctial neap tides: and those of Colepresse, on the tides at Plymouth, which give 16 feet for the mean height (intermediate between spring and neap) and 9 feet difference between springs and neaps. Preferring the proportion deduced from the former, he considers the height of equinoctial spring tides to be to that of equinoctial neap tides as 9:5. These tides (as will be seen hereafter) are in one case the effect of the moon augmented by the effect of the sun; and in the other case the effect of the moon diminished by that of the sun. If no correction were needed, we should infer at once that the power of the moon is to that of the sun as 7:2. But Newton remarks, that the greatest tides at Bristol do not happen till 43 hours after syzygies, "ob aquarum reciprocos motus," meaning, probably, that the oscillations, like the oscillations of a pendulum, have a kind of inertia, which (on purely mechanical principles) prevents them from attaining their greatest magnitude till the force which causes them has past its greatest magnitude. This we shall find, when we treat of Waves, to be incorrect, except we take account of friction. Assuming this, however, Newton proceeds Tiles and to correct for the position of the luminaries at the Wares. instant of Bristol high tide: remarking that, as the sun is 184 degrees from the moon at spring tides, and Set. II. 90°+181° at neap tides, it is not the whole force of the sun which in one case increases and in the other of case diminishes the moon's effect, but the whole force of the sun x cos 37°: and also that, as the moon's declination, 43 hours after an equinoctial syzygy, is about 22°, it is not the whole force of the moon that is ron concerned, but the whole force of the moon x cos 22°. These corrections appear to us inconsistent with what has gone before: for if the tides are increasing from the accumulated action of the sun and moon during a long time, it seems clearly inaccurate to correct the results of observation for the places of those bodies at the very instant of observation. Then he observes that the moon is not, at syzygies, at her mean distance. All corrections applied, he finds that the force of the moon is to that of the sun as 4 4815 to 1: and, therefore, as the sun's force would raise the water 1 foot, 11 inches, the moon's force would raise it 8 feet, 8 inches. This, he remarks, is amply sufficient to account for all the motions of the tides. (18.) The proportion of the moon's tidal force to the sun's tidal force is used by Newton (as a different value found in nearly the same manner has been used by Laplace) as the basis on which he calculates the moon's mass for application to other parts of the theory of gravitation. We shall see grounds hereafter for questioning the propriety of this calculation. (19.) Assuming that Newton intended here (as he has done in several parts of Optics) only to exhibit, as far as he was able, grounds for a numerical calculation relating to the subject of Tides, but not bearing directly upon any of its specific phænomena, we must allow that (in spite of the apparent inconsistency of his corrections) it is a wonderful first attempt. That it had no further meaning will be sufficiently evident, not only from the proposition already cited, lib. I., prop. 66, cor. 19, but also from an examination of his 24th proposition of the third book, and the first corollary of his 27th proposition. In these he has treated the geneal explanation of the Tides as a matter of Wave-theory tirely, (though not without errors,) particularly in regard to the interference of semidiurnal tides, and in explaining the small rise and fall at some islands in the open sea by the oscillation of the whole mass of water between the bounding continents. As a philosopher, we conceive Newton to have shown himself here superior to his successors. (20.) In explaining the more complete equilibriumtheory, we shall not confine ourselves to the methods of Daniel Bernoulli, or any other writer, but shall present the theory in the form which appears most convenient. The problem which we shall conceive to be presented to us for solution is this: suppose the earth to be a bem of spherical solid nucleus, either homogeneous, or conequili sisting of a series of spherical concentric strata, (each stratum having the same density and the same thickness in its whole extent,) which nucleus is covered with water and suppose the disturbing forces of the sun and moon to act upon the water to find the shape which the water will assume. 1 same on a (21.) We have designedly used the word spherical Tides and for the form of the earth, because the investigation of Waves. the alteration produced in the form which, if undis- Tides the turbed, would be spheroidal, would prove rather troublesome, and would lead to no result which we spherical shall not obtain without it. As the earth's ellipticity earth as on is small, (the difference between its major axis and its a spheroid. minor axis being only about of either,) and as the 300 whole elevation of the water, on the equilibriumtheory, is but a few feet, the reader will have no difficulty in comprehending that the tidal elevation of the water on the spheroid, though without doubt theoretically different from that on a sphere, will practically differ by a quantity which is quite insensible. In the same manner the reader will understand The tide that, supposing the water to be disturbed by the action produced of the sun, and supposing the action of the moon to by each of be then introduced, the additional disturbance which it will cause will be (as far as the senses can discover) ries the ing lumina. the same as it would have caused if it had acted on same as if water not disturbed by the action of the sun. And the other did not thus the whole disturbance which the two luminaries will produce upon the water surrounding a spheroidal nucleus will be found with sufficient accuracy by investigating the disturbance which each of them, separately considered, would produce in the water surrounding a spherical nucleus, and by adding those two disturbances together. (22.) Our first effort will now be directed to the estimation of the disturbing force of the sun upon the water. We shall use the following notation :— K, the mean density of the earth's spherical nucleus : R, its radius. k, the density of the water: r, the radius of the external spherical surface of the water when undisturbed by the sun and moon. (The density is supposed to be estimated by the acceleration which a cubical unit of matter acting by its attraction during a unit of time will produce in a body whose distance is the unit of distance: the velocity and acceleration being referred to the same units.) E, the whole mass of the earth and water. 9, the numerical expression, referred to the same units, for the acceleration which gravity at the earth's surface causes in bodies falling freely. x, y, z, the rectangular coordinates of any point in the fluid, the centre of the spherical nucleus being the origin, and being parallel to the line joining the centres of the sun and the earth. D, the sun's distance: Dm, the sun's mean distance: P, the sun's parallax: Pm, the sun's mean parallax: T, the periodic time of the earth's revolution round the sun, or the length of a sidereal year: S, the sun's mass, estimated by the acceleration which it will produce (in the same manner as for the density, above). D', the moon's distance: D'm, the moon's mean distance: P', the moon's parallax: P', the moon's mean parallax: T', the periodic time of the moon's revolution round the earth: M, the moon's mass. the attract exist. Actual forces of (23.) The distance of the sun from the point whose co-ordinates are x, y, z, is √{'+y+(D-2)}, and the the Sun attraction of the sun upon that point, according to the law of gravitation, is S x2+y2+(D-2) This force is in the upon any particle of the water. Waves. Waves Tides and direction of the line drawn from the point in question to the sun. Our expression for this force supposes it to be Tides ar estimated as an accelerating force; the statical pressure which corresponds to it may be resolved into three pressures in the directions of x, y, z; and, by the principle that accelerations of a given particle are proportional to Sect. I the pressures which cause them, the accelerating forces which act in these directions may be deduced from the Equili given accelerating force by the same laws of resolution as those for statical pressures. Thus we find for the briumresolved parts of the sun's accelerating force on the particle in question, Expansions of the expres sions. 2 Theory Tides. (24.) Now the proportion of the earth's radius to the distance of the sun is extremely small; and the value of x y or is necessarily smaller. It will be allowable, therefore, to expand these expressions approximately, D'D' D' retaining no higher powers of x, y, z, than the second. (Indeed these latter terms are wholly insensible for the sun; and we retain them only because, in the expressions which we shall infer by analogy for the forces of the moon, they may be considered sensible.) With this restriction, observing that Disturbing forces of the Sun upon every particle. (25.) These expressions represent the whole force of the sun upon any particle. But it is evident that, to find the force which disturbs the form of the water in reference to the position of the earth, we must not use the whole force of the sun upon any particle, but the excess of the sun's force on the particle above the sun's force on the centre of gravity of the earth. In order to find the sun's force on the centre of gravity of the earth, we must multiply each particle of the earth by the force which acts upon it; we must add together all these products, and we must divide the sum by the sum of all the particles of the earth. Now, using the 'expressions above, (which apply to the earth as well as to the water,) we may easily see that, if we multiply each particle D3 . - Sr of the earth by the force and add all the products together, the sum will be 0, because for every particle which has a certain positive value of a there will be another particle having an equal negative value of x, and their products will, when added, destroy each other. The same remark applies to the terms depending on y, z, S x2, and yz. But it does not apply to the term or to that depending on x2, y2, and 2o. S (26.) Now for the term we have only to remark that, upon multiplying it by each of the particles, adding D2 S all the products, and dividing the sum by the sum of the particles, we again obtain For the other terms we D may proceed thus:-The sum of all the products of each particle by its value of 22, throughout the sphere, will be the same as the sum of the products of each particle by its value of or y3, because, supposing the sphere at one time divided by planes perpendicular to z, and at another time by planes perpendicular to x or y, the sections for similar values of x, y, or z, will be similar. The sum, therefore, for 62 will be equal to that for 3x2+ 3y', and, therefore, that for 622-3x2-3y2 will be 0. The only remaining term, therefore, for the sun's force on the centre of gravity of the earth, is in the direction of z. S (27.) Subtracting this term, therefore, from the force in the same direction upon the particle under consideration, we have the following expressions for the sun's disturbing force, Sect. II. dU dy dU posed in tion which the form of (28.) We shall now proceed to investigate the form which the water covering the solid nucleus will receive from The den the action of these forces in addition to the attraction of the nucleus and the mutual attraction of the particles of sity of the of water. And first we may remark that, if the attraction of the particles of the water is insensible, (or if the den- fluid supsity of the water is insensible in comparison with that of the nucleus,) the problem is very simple. Referring to significant. our Treatise on the FIGURE OF THE EARTH, section 2, article 7., we find that the condition for the possibility of Mathemaequilibrium of the water is that Xdx+Ydy+Zdz shall be a complete differential, or, in more correct language, tical condidU that it shall be possible for us to find some function U, such that =X, =Z; X, Y, Z, being the determines dx whole forces in the directions of x, y, z. In article 9. of the same Treatise it is shown that, when the forces are the fluid produced by attraction to any number of particles, this condition is always satisfied; and, therefore, it is satis- when in fied here (which will also be easily seen on substituting the expressions which we shall immediately exhibit). equiliIn article S. of the same Treatise, it is proved that the form of the external surface will be determined by making Xdx+Ydy+Zdz=0, or U=C. To apply this now, we must add, to the expressions above, the resolved parts of the attraction of the nucleus. That attraction is the same as if all the matter of the nucleus were collected at dz Hence the whole forces acting on any particle of the water are Sr 3Sr 3 (x2+ y2+z2)? D3 D R3K2 3 ° (x2 + y2+z2)}* brium. dr dU dy We may remark, that the very same equation would have been obtained if we had considered only the disarbing force which acts in the direction of a tangent to the Earth's surface. For, the equation which we have used for the external surface amounts to this, that the whole force is perpendicular to the external surface. Therefore the inclination of the surface of the water to the surface of the sphere will depend entirely on the proportion of the tangential force to the force directed towards the centre of the sphere. The only tangential force is the tangential disturbing force, which must therefore be retained; but the force directed to the centre of the sphere consists of the attraction of the sphere and the minute disturbing force; and it is indifferent, for the inclination of which we have spoken, whether we retain that minute portion or not. If we retain it, we consider all the forces; if we omit it, we use no disturbing force but that which is tangential. We shall see hereafter that a similar rule is true when we consider the forces producing the motion of the sea. the form to (29.) Since the difference of the form from a spherical form will be exceedingly small, we may for Expansion (*+ y2+z2), which is the distance of any point at the surface from the sphere's centre, put r+q (then q is the supposing elevation of the water above the height which it would have had if undisturbed by the attraction of the sun): differ little and in substituting this expression in the first term on the right hand side of the equation we may neglect the from a square of q; and in substituting it in the factors of the other terms, which are exceedingly small, we may omit sphere q entirely. In these small terms, therefore, |