of OOl^NOfc® 

 L 139 ] 



IV. 



THE CENTEE OF GRAVITY AND THE PEINCIPAL AXES OF 

 ANY SUEFACE OF EQUAL PEESSUEE IN A HETEEOGENEOUS 

 LIQUID COVEEING A HETEEOGENEOUS SOLID COMPOSED 

 OF NEAELY SPHEEICAL SHELLS OF EQUAL DENSITY, WHEN 

 THE WHOLE MASS IS EOTATING WITH A SMALL ANGULAE 

 VELOCITY IN EELATIVE EQUILIBEIUM UNDEE ITS OWN 

 ATTEACTION. 



By M. W. J. FEY, M.A, F.T.C.D. 



Read June 24. Ordered for Publication June 26. Published December 27, 1907. 



In his " Mecanique Celeste," Livre iii., chap, iv., Laplace discusses, on certain 

 assumptions, the forms of the surfaces of equal pressure or density in a 

 heterogeneous liquid covering a heterogeneous solid earth, when the whole is 

 rotating with a small uniform angular velocity in relative equilibrium under 

 its own attraction and that of distant spherical bodies. The assumptions are : 

 that the earth is composed of almost spherical shells of equal density ; that 

 the surfaces of equal density or pressure in the liquid are almost spherical ; 

 and further (although Laplace does not state so), that either the distant 

 bodies rotate round the same axis as the earth and with the same angular 

 velocity, or that, for a first approximation, it is possible to neglect any accele- 

 ration which a particle of the liquid must have additional to that due to the 

 angular velocity, during the motion of the liquid as it adapts itself to the 

 varying form it must assume owing to the rotation of the earth relatively to 

 the distant bodies. 



Expressing the radius vector r to any point on a surface of equal density 

 in the form r = a + aa (Yi + Y^ + &c.), where Y^, Y2, &c., are spherical 

 surface-harmonics, a is a small constant whose square can be neglected 

 and a is the radius of the sphere of equal volume, Laplace shows that 

 Yn must satisfy the following condition at any point of a surface of equal 

 density or pressure in the liquid 



' )rfo"+^ Yn — - x^Y.+ l pel -^^ + — Zn - 0, 



where xf. 



pa~da, and a is the value of a at the free surface. Hence it 



follows that any one of the 2n + 1 constituents of F„, consisting as they do 

 p, I. A. pr.oc, VOL. yxvii., sect. a. • [20] 



