LAPLACE 303
suffer from his severe application. He married a beautiful and amiable woman, and left a son, born in 1789, who succeeded to his title, and rose to the rank of general in the artillery.
It might be said that Laplace was a great mathematician by the original structure of his mind, and became a great discoverer through the sentiment which animated it. The regulated and persistent enthusiasm with which he regarded the system of nature was with him from first to last. It can be traced in his earliest essay, and it dictated the ravings of his final illness. By it his extraordinary analytical powers became strictly subordinated to physical investigations. To this lofty quality of mind he added a rare sagacity in perceiving analogies, and in detecting the new truths that lay concealed in his formulae, and a tenacity of mental grip, by which problems, once seized, were held fast, year after year, until they yielded up their solutions. In every branch of physical astronomy, accordingly, deep traces of his work are visible. "He would have completed the science of the skies," Fourier remarks, "had the science been capable of completion."
For a fuller account of the results achieved by him, the article
ASTRONOMY, vol. ii. p. 761, may be consulted ; it need only be added
that he first examined the conditions of stability of the system
formed by Saturn's rings, pointed out the necessity for their rotation, and fixed for it a period (10 h 33 m ) differing by little more than a minute from that established by the observations of Herschel; that he detected the existence in the solar system of an invariable plane such that the sum of the products of the planetary masses by the projections upon it of the areas described by their radii vectores in a given time is always a maximum, made notable advances in the theory of astronomical refraction (Méc. Cél., tom. iv. p. 258), and constructed formulae, agreeing remarkably with observation, for the barometrical determination of heights (Méc. Cél., tom. iv. p. 324). His removal of the considerable discrepancy between the actual and Newtonian velocities of sound,[1] by taking into account the increase of elasticity due to the heat of compression, would alone have sufficed to illustrate a lesser name. Molecular physics also engaged a large share of his attention, and he announced in 1824 his purpose of treating the subject in a separate work. With Lavoisier he made an important series of experiments on specific heat (1782-84;, in the course of which the "ice calorimeter" was discovered; and they contributed jointly to the Mémoirs of the Academy (1781) a paper on the development of electricity by evaporation. Laplace was, moreover, the first to offer a complete analysis of capillary action based upon a definite hypothesis that of forces "sensible only at insensible distances"; and he made strenuous but unsuccessful efforts to explain the phenomena of light on an identical principle. It was a favourite idea of his that chemical affinity and capillary attraction would eventually be included under the same law, and it was perhaps as much because it threatened an inroad on a cherished generalization as because it seemed to him little capable of mathematical treatment that the undulatory theory of light was distasteful to him.
The investigation of the figure of equilibrium of a rotating fluid mass engaged the attention of Laplace during the greater part of his long life. His first memoir was communicated to the Academy in 1773, when he was only twenty-four years of age, his last in 1817, when he was sixty-eight. The results of his many papers on this subject – characterized by him as "un des points les plus interessans du système du monde" – are embodied in the Mécanique Céleste, and furnish one of the most remarkable proofs of his analytical genius. Maclaurin, Legendre, and D'Alembert had furnished partial solutions of the problem, confining their attention to the possible figures which would satisfy the conditions of equilibrium. Laplace treated the subject from the point of view of the gradual aggregation and cooling of a mass of matter, and demonstrated that the form which such a mass would ultimately assume must be an ellipsoid of revolution whose equator was determined by the primitive plane of maximum areas.
The honour of having brought almost to perfection the closely related problem of the attraction of spheroids must also be accorded to him. All the powers of analysis in the hands of its greatest masters replaced the old geometrical methods, and their superiority was soon evidenced by a succession of remarkable discoveries. Legendre, in 1783, extended Maclaurin's theorem concerning ellipsoids of revolution to the case of any spheroid of revolution where the attracted point, instead of being limited to the axis or equator, occupied any position in space; and Laplace, in his treatise Théorie du Mouvement et de la Figure Elliptique des Planètes (pub-
1
lished in 1784), effected a still further generalization by proving, what had been suspected by Legendre, that the theorem was equally true for any confocal ellipsoids. Finally, in a celebrated memoir, Tluorie des Attractions des Spheroides et de la Figure des Planetes, published in 1785 among the Paris Memoirs for the year 1782, written, however, after the treatise of 1784, Laplace treated exhaustively the general problem of the attraction of any spheroid upon a particle situated outside or upon its surface. The researches of Laplace and Legendre on the subject of attrac tions derive additional interest and importance from having intro duced two powerful engines of analysis for the treatment of physical problems, Laplace's Coefficients and the Potential Function. The expressions for the attraction of an ellipsoid involved integrations which presented insuperable difficulties; it was, therefore, with pardonable exultation that Laplace announced his discovery that the attracting force in any direction could be obtained by the direct process of differentiating a single function. He thereby translated the forces of nature into the language of analysis, and laid the foundations of the mathematical sciences of heat, electricity, and magnetism. This function, V, which received the name of potential from Green in 1828, and independently from Gauss in 1840, is defined as the sum of the masses of the molecules of the attracting body divided by their respective distances from the attracted point ; or, in mathematical language pdxdydz "-fffw=. p being the density of the body at the point #, y, z ; a, /3, y, the coordinates of the attracted point ; and the limits of integration being determined by the form of the attracting mass. V is thus a function of a, /3, 7, that is to say, depends for its value on the position of the point, and its several differentials with respect to these coordinates furnish the components of the attractive force. The integrations, however, could not in general be effected so as to express V in finite terms; but Laplace showed that V satisfied the partial differential equation
- V *V #7
da 2 dp-dyt
which is still known as Laplace's equation. It is worthy of remark that it was not in this symmetrical form that the equation was discovered, but in the complicated shape which it assumes when expressed in polar coordinates: –
where /j. is substituted for cos 0. This differential equation forms
the basis of all Laplace's researches in attractions, and makes its
appearance in every branch of physical science.
The expressions which are known as Laplace's coefficients, a name first given to them by Dr Whewell, 2 occupy a distinguished place in modern analysis. They were first introduced in their generality by Laplace in the memoir on attractions, 1785, above referred to, which is, to a great extent, reprinted in the third book of the Mécanique Céleste; but Legendre, in a celebrated paper entitled Recherches sur l'attraction des Sphéroides homogènes, printed in the tenth volume of the Divers Savans, 1783, had previously made use of them, and proved some of their properties, in the simplified form which they assume with one instead of two variables. They may be defined as follows. If two points in space are determined by their polar coordinates, r, 6, w, and r , & , ca , T the reciprocal of the distance between them is expressed in terms of those coordinates by
>" - 2) u + V 1 fj? Vl - p" cos (co-o/)f +/ 2 T
where /j. and a are written for cos 9 and cos 6 respectively. This expression may be expanded in a series of the form
T J. i~^T T i a y/2 "*" i * r i ^
where P , 1 . . . P,- are Laplace's coefficients of the orders 0, 1 . . . i. They are rational integral functions of /u, /l - /u 2 cos u, and / - /it 2 sin u, and are precisely the same functions of /u , VI - /it 2 cos <a and VI a 2 sin &>,; or, in other words, of the rect angular co-ordinates of the two points divided by their distances from the origin. The general coefficient P,- is of i dimensions in these quantities, and its maximum value can be shown to be unity, so that the above written series will be convergent if r is greater 2 See Monthly Notices of the Astronomical Society, xxvii. p. 211. They are also included in the more general expression " Spherical harmonics" (" Fouctions spheriques," " Kugelfunctionen ").
- ↑ Annales de Chimie et de Physique, 1816, tom. iii. p. 238.
