The resulting equation being found to correspond with the general equation to a spheroid, a comparison of like terms in the two equations gives the values of the constants which enter into the former. If r represent the mean distance of the spheroidal surface of the water from the centre of the earth, and + lt represent the distance of any point on that surface above or below the mean level ; then at the surface ; and the determination of h for any place gives at that place the height of the water above, or its depression below the mean level.
Uniting the effects of the solar and lunar disturbances by simply adding them together, since the disturbing forces are very small com pared with the force of gravity ; and introducing, in place of the rectangular coordinates, angles which depend on the longitude and latitude of a station, with the right ascension and declination of the sun and moon, the value of the term ± h may be shown to consist of three parts : one of these depends on the variation of the declination of the sun and moon, and indicates a slow tide which goes through its changes in about fourteen days; the second depends on the hour angles both of the sun and moon, and iudicates two tides which go through their changes in a solar and a lunar (lay respectively. These being combined, there is produced a diurnal tide, the highest state of which should precede, at a variable interval, the moon's culmination between the times of passing from syzygy to quadrature, and should follow it between the quadratures and syzygies. It has been found, however, that the observed accelerations and retardations, and also the absolute elevations of the water, in very few cases agree with the results of the theory. [ACCELERATION AND RETARDATION OF TIDF.S.] The third part depends upon the doubles of the hour angles just mentioned, and consequently indicates two semi-diurnal tides, which being combined constitute one such tide, whose highest state is variable. The nature of the expression shows that the semi-diurnal tide should be the greatest at the equator, and should diminish till it vanishes at the poles : it denotes also that it is greatest at new or full moon, and least at the quadratures. The theory moreover indicates that the difference between two consecutive tides ought to be very considerable in Europe; whereas they are known to be nearly equal to one another. Both Newton and Bernoulli endeavoured to explain this circumstance by the hypothesis of a general oscillation of the sea, in consequence of which the highest tide gives to the lowest a quautity equal to the difference between them ; hut the researches of La Place have shown that, even with such oscillations, the two tides could not (according to the theory) be equal unless the sea were everywhere equally deep.
Euler, departing from the hypothesis that the.sea is always in equi librio under the action of the sun and moon, endeavoured to introduce the subject of fluid oscillations in his theory of the tides; but the laws of undulation were not then known, and Euler assumed that a molecule of the sea in motion endeavours to regain the position which, in a state of equilibrium, it would occupy in a vertical line with a force proportional to its vertical distance from that position.
The theory adopted by La Place, in which there are taken into con sideration the laws of the motion of fluid molecules when acted on by attracting forces, was a great improvement on that of the mathema ticians before mentioned ; and it is found to produce a more near agreement with the observed phenomena. The elaborate investigations of La Place will be found in the Mdmoires de l'Acaddmie des Sciences' for the years 1775, 1776 ; and in the first and fourth books of the Mdcanique Celeste.' They are also given, so far as contained in the first book, in the late Dr. Thomas Young's Elementary Illustrations of the Celestial Mechanics of Laplace ; ' Lend.,1821. As in the former theory, the solid nucleus of the earth is supposed to be entirely covered with water of uniform depth ; and the investigations com mence with the proof (` 316c. liv. i., eh. 8) that any portion of the water, however its place may bo changed, will always retain the same volume. The equation expressing this law is called the equation of continuity.
A very small parallelopiped of water within that which covers the solid nucleus of the earth is acted upon by accelerative forces arising from pressures estimated in the directions of three rectangular coordinate axes whose origin is at the centre of the earth : the first is supposed to be parallel to the axis of rotation, and the others in the plane of the equator: one being directed to the equinoctial point and the other at right angles to that direction. The pressures are supposed to arise from the attraction of the earth, from the angular velocity of its rota tion, and from the disturbing forces, and to tend towards the origin of the coordinates.