ON THE PHYSICAL STRUCTURE OF THE EARTH. 209 



As precession depends essentially on the variation of the angle //', it 



G — A • 



follows that the complete expression of the ftictor — ^ — is of primary 



importance. 



(4) Mathematicians, dnring tlie past two centnries, have devoted 

 much attention to the question of the figure of a rotating mass of fluid, 

 with esi)ecial reference to the explanation of the spheroidal figures of 

 the earth and her sister planets. Solutions of this problem have been 

 presented, especially by Clairaut, Legendre, Laplace, Gauss, Ivory, 

 Jacobi, and Airy ; and it is not a little remarkable that in applying 

 these solutions to the case of the Earth every one of these investi- 

 gators has not only supposed the Earth to have been originally in a 

 fluid state, but that the particles of the mass retained the same posi- 

 tions after solidification had taken place. This tacit or openly expressed 

 assumption of the unchangeable position of the particles of the origi- 

 nal fluid mass on their passage to a complete or partial state of solidity 

 lies at the root of the whole question of the Earth's structure. For the 

 first time in the treatment of the physico-mathematical problem, I dis- 

 tinctly discarded this assumption and I affirmed that the position of 

 the particles of matter, on passing from the state of fluidity to solid- 

 ity, must assume positions in conformity with mechanical and physical 

 laws. In this way the hypothesis of the Earth's primitive fluidity be- 

 came more simple and much more rational ; for it was as manifestly 

 absurd to assume that the particles of the fluid mass, on passing into 

 a solid state of consistence, retained their original positions, as it would 

 be to assume that if the whole Earth became liquefied the positions of 

 its particles would be unchanged. The corrected and simplified 

 hypothesis is also fruitful in important results ; but it is singular that, 

 as far as I am aware, no mathemetician seems to have understootl or 

 appreciated its bearing on the physical structure of the Earth, except 

 M. Plana, by a remark iu a memoir published by him towards the close 

 of his career. 



(5) Before presenting my conclusions on the shape of the inner surface 

 of the solidified shell and Plana's remark relative to the same subject, 

 it is necessary to recall some results established by Clairaut and fre- 

 quently put forward by mathematical investigators of the Earth's figure. 

 It seems to be universally admitted that if a mass of heterogeneous fluid 

 composed of strata of equal density, each increasing iu density from the 

 surface of the mass to its center, is set in rotation, the several strata 

 will be spheroidal, but their ellipticities will not be equal. The elliptici- 

 ties will decrease from the outer surface toward the center. This law 

 of decrease of ellipticity toward the center is not a hypothetical residt, 

 but a necessary deduction from the properties of fluids. As all known 

 fluids are compressible, such an arrangement of strata of equal density 

 as that referred to must follow from the supposition of the existence of 

 any mass of fluid of such magnitude as the whole Earth. The increase 



H. Mis. 129 14 



