Another discovery relative to the constitution of the planetary system, which does infinite honour to the sagacity of Laplace, is the cause of the secular inequalities indicated by ancient and modern observations in the mean motions of Jupiter and Saturn. On examining the differential equations of the motions of these planets, Laplace remarked, that as their mean motions are nearly commensurable (five times the mean motion of Saturn being nearly equal to twice that of Jupiter), those terms of which the arguments are five times the mean longitude of Saturn, minus twice that of Jupiter, may become very sensible by integration, although multiplied by the cubes and products of three dimensions of the eccentricities and inclinations of the orbits. The result of a laborious calculation confirmed his conjecture, and showed him that in the mean motion of Saturn there existed a great inequality, amounting at its maximum to 48 2" 3, and of which the period is 929 years ; and that in the case of Jupiter there exists a corre sponding inequality of nearly the same period, of which the maximum value is 19 46", but which is affected by a contrary sign, that is to say, it diminishes while the first increases, and vice versa. He also perceived that the magnitude of the co-efficients of these inequalities, and the duration of their periods, are not always the same, but participate in the secular variations of the elements of the orbit.
The theory of the figures of the planets, scarcely less interesting than that of their motions, was also greatly advanced by the researches of Laplace. He confirmed the results of Clairaut, Maclaurin, and D Alembert, relative to the figure of the earth, and treated the question in a much more general way than had been done by those three great mathematicians. From two lunar inequalities depending on the non-sphericity of the earth, he determined the ellip- ticity of the meridian to be ^^ T very nearly.
Newton, in the Principia, explained the cause of the phenomena of the tides, and laid the foundations of a theory which was prosecuted and extended by Daniel Bernouilli, Maclaurin, Euler, and D Alembert ; but as none of these geometers had taken into account the effects of the rotatory motion of the earth, the subject was in a great measure new when it was taken up by Laplace in 1774. Aided by D Alembert s recent discovery of the cal culus of partial differences, and by an improved theory of hydrodynamics, he succeeded in obtaining the differential equations of the motion of the fluids which surround the earth, having regard to all the forces by which these motions are produced or modified, and published them in the memoirs of the academy in 1775. By a careful examina tion of these equations, he was led to the curious remark, that the differences between the heights of two consecutive tides about the time of the solstices, as indicated by New ton s theory, are not owing, as Newton and his successors had supposed, to the inertia of the waters of the ocean, but depend on a totally different cause, namely, the law of the depth of the sea, and that it would disappear entirely if the sea were of a uniform and constant depth. He also arrived at the important conclusion, that the fluidity of the sea has no influence on the motions of the terrestrial axis, which are exactly the same as they would be if the sea formed a solid mass with the earth. The same analysis conducted him to the knowledge of the conditions necessary to ensure the permanent equilibrium of the waters of the ocean. He found that if the mean density of the earth exceeds that of the sea, the fluid, deranged by any causes whatever, from its state of equilibrium, will never depart from that state but by very small quantities. It follows from this, that, since the mean density of the earth is known to be about five times greater than that of the sea, ths great changes which have taken place in the relative situation of the waters and dry land must be referred to other causes than the instability of the equilibrium of the ocean.
The chief steps in the progress of the study of tides since the time of Laplace have consisted in co-ordinating the- results of observation, and analysing them into their partial phenomena, by the help of Newton s and Bernouillrs theory. This labour has been greatly advanced by Dr TVhewell, and also by Sir John Lubbock. The former has con structed maps of " cotidal lines," which, indicating the relative time of high water in different parts of the globe, give us a graphic conception of the course and propagation of the tidal wave. The tides of the Eastern Pacific are but little known ; but a vast wave advances northwards between Australia and Africa, diverted or retarded by the obstacles it meets with in the Indian Archipelago. Another (and to us the most important) branch sets from south to north up the vast canal of the Atlantic, where it is gradually complicated by local tides, having their origin in the wide expanse between Africa and the Gulf of Mexico. The two sets of waves sometimes reinforce, sometimes oppose one another ; they are prolonged to the western shores of England and Norway, where the tidal impulse arrives twenty-four hours after it passed the Cape of Good Hope. It is propagated most rapidly at a distance from coasts, and is retarded in narrows and shallows. It sends offshoots into every bay and strait, always greatly retarded in point of time (apparently by friction), but often increased in elevation by concentration of the effect in a gradually narrowing channel, as we see in the exaggerated tides of the river Amazon, the Severn, and the Bay of Fundy. The same place may be the seat of several tides at once, which may increase or destroy one another ; thus, a small tide is propagated through the Straits of Dover as far as the Dutch coast, where it only arrives simultaneously with the principal wave, which has made the entire tour of Great Britain.
As regards the progress of theory, Dr Thomas Young, one of the greatest philosophers of this century, next after Laplace grappled with the difficulties of this arduous subject. Employing mathematical methods of inferior power but greater directness, and taking into account causes of local action which Laplace had not ventured to include in his analysis, he gradually matured a theory adequate to represent many of the results of experience, of which Laplace gives no account. He distinguishes the results of the forced and free oscillations of the sea : the former resulting from the direct action of the sun and moon combined with the rotation of the earth, and whose periods of rise and fall are determined solely by those external causes ; the free waves, on the contrary, derived from the former, are transmitted with velocities depending on the mechanism of the fluid itself, on its depth, and on the resistances arising from friction to which those motions are exposed. These all-important modifications of the dynamical theory of the tides were deduced by Young from the general theory of oscillations and resistances, and from the laws of fluids detected by Du Buat, and he applied them with no ordinary skill to the solution of the problems of tides in oceans, estuaries, and rivers. It is satisfactory to find that by an independent and very different method Airy subsequently arrived at substantially the same results as Young.
of the precession of the equinoxes, which also received similar improvements in passing through the hands of Laplace. He demonstrated, as has been mentioned, that the fluidity of the sea has no influence on the phenomena of precession and nutation. He considered some of the
effects of the oblate figure of the earth which had not been