158 



Prof. G. H. Darwin. 



[Dec. 13, 



IV. " On the Figure of Equilibrium of a Planet of Heterogeneous 

 Density." By G. H. Darwin, F.R.S., Plumian Professor of 

 Astronomy and Fellow of Trinity College, Cambridge. 

 Received December 3, 1883. 



The problem of the figure of the earth has, as far as I know, only 

 received one solution, namely, that of Laplace.* His solution involves 

 an hypothesis as to the law of compressibility of the matter forming- 

 the planet, and a solution involving another law of compressibility 

 seems of some interest, even although the results are not perhaps so 

 conformable to the observed facts with regard to the earth as those of 

 Laplace. f 



The solution offered below was arrived at by an inverse method, 

 namely, by the assumption of a form for the law of the internal 

 density of the planet, and the subsequent determination of the law of 

 compressibility. One case of the solution gives us constant compres- 

 sibility, and another gives the case where the modulus of compressibility 

 varies as the density, as with gas. 



It would be easy to fabricate any number of distributions of 

 density, any one of which would lead to a law of compressibility 

 equally probable with that of Laplace ; but the solution of Clairaut's 

 equation for the ellipticity of the internal strata of equal density 

 seems in most cases very difficult. Indeed, it is probable that Laplace 

 formulated his law because it made the equation in question integrable, 

 and because it was not improbable from a physical point of view. 



The following notation will be adopted : — 



For an internal stratum of equal density let — 



r be the radius vector of any point, 

 a the mean radius of the stratum, 

 e the ellipticity, 

 w the density, 



colatitude from the axis of rotation, 



p the hydrostatic pressure at the point r, 0. 



For the surface let r, a, t } io denote the similar things. 

 Let M be the mass of the planet, p its mean density, iv the angular 

 velocity of rotation, m the ratio w i \^7rp. 



* Since this paper was presented I have seen a reference to a paper by the late- 

 M. Edotiard Koche, in vol. i of the Memoirs of the Academy of Montpellier (1848), 

 in which the problem is solved, when the rate of increase of the density varies as 

 the square of the radius. See Tisserand, " Comptes Rendus," 23rd April, 1883. 



f Laplace's hypothetical law of compressibility arises from a law of internal 

 density for which the problem had previously been worked out, as an example, by 

 Legendre. See Todhunter's " History of the Figure of the Earth," vol. ii, pp. 117 

 and 337. 



