CHAP. X.

Of the Tides.

Ir the investigation of the laws of the equilibrium of the fluids which cover the planets, presents great difficulties, that of the motion of these fluids agitated by the attractions of the heavenly bodies, offer still greater.

Thus Newton, who occupied himself the first with this important problem, was satisfied with determining the figure in which the ocean would remain in equilibrio, under the action of the Sun and Moon. He supposed that the sea, at every instant, took this figure; and this hypothesis, which extremely facilitates the calculations, gave him results, in many respects conformable with the observations. In fact, this great geometrician had re

course to the action of the rotation of the Earth, to explain the retardation of the tides, beyond the passage of the Sun and Moon over the meridian; but his reasoning is unsatisfactory, and, moreover, appears contrary to the result of a rigorous analysis. The Academy of Sciences proposed this subject for a prize, in 1740; the successful pieces, contained the developement of the Newtonian theory, founded on the same hypothesis, of the ocean in equilibrium under the action of the attracting bodies. It is evident, nevertheless, that the rapidity of the Earth's motion prevents the waters that cover it, from taking at every instant, the figure suitable to the equilibrium of the forces, but the investigation of this motion, combined with that of the action of the Sun and Moon, was too difficult to be effected by the state of analysis at that time, and of the knowledge then possessed of the motions of fluids. But assisted by the discoveries which have since been made on both these subjects, I have again undertaken this

problem, the most intricate in celestial mechanics. The only hypotheses which I shall permit myself are, that the ocean inundates the whole Earth, and that it meets with but slight obstructions in its motion; the rest of my theory is rigorously exact, and founded on the principles of the motion of fluids. By thus conforming to nature, I have the satisfaction to see my results agree with the observations, particularly with respect to the small difference which subsists between the two tides of one day, which difference, according to the theory of Newton, should be very great. I obtained this remarkable result, namely, that to make this difference disappear, it is only necessary to suppose the ocean to have every where the same depth. Daniel Bernoulli, in his Essay on the Tides, which divided the prize of the academy, in 1740, endeavoured to explain this phenomenon, by supposing that the motion of the Earth was too rapid to permit the tides to accommodate themselves to the theory. But it can be shewn by analysis,

that this rapidity could not prevent the tides from being very unequal, if the depth of the ocean was not constant. We may see by this example, and by that of Newton, how much we should distrust the most plausible hypotheses, when not supported by rigorous calculation.

The preceding results, though very extensive, are still restricted by the supposition of a fluid, regularly spread over the Earth, and subject to very slight resistances in its motions. The irregularity in the depth of the ocean, the position and declivity of the shores, their situation relative to the neighbouring coasts, the friction of the waters against the bottom of the ocean, and the resistances they meet with all these causes, which it is impossible to reduce to calculation, modify the oscillations of this great fluid mass. All that can be done is to analyse the general phenomena of the tides, which should result from the attractive forces of the Sun and Moon, and to derive from observation such data as are indispensably

necessary to complete the theory of the tides for each particular port. These data are so many arbitrary quantities depending on the extent of the sea, its depth, and the local circumstances of the port. Under this point of view, we shall consider the oscillations of the ocean, and their correspondence with observations.

Let us first consider the action of the Sun alone upon the ocean, and suppose its motion uniform in the plane of the equator. It is evident, that if the Sun acted on the centre of gravity of the Earth, and of every particle of the ocean, by exerting equal and parallel forces, the whole system of the terrestrial spheroid would obey these forces by a common motion, and the equilibrium of the waters would not be at all altered. This equilibrium, then, is only deranged by the difference of these forces, and by the inequality of their directions. A particle of the ocean, placed directly under the Sun, is more attracted than the centre of the Earth. It tends,