to Observed Latitudes. 303 



semiaxes arc c and t'(l + e), e being a very small quantity 

 whose square may be neglected, at an external point whose 

 coordinates are/, 0, li, is 



^^'^{^^l^f-m}, 



where M is the mass of the spheroid ; if p be the density, 



M = f7rpcXl + 2e). 



If the spheroid, instead of having a uniform density p, be 

 formed of homogeneous spheroidal shells having a common 

 centre and axis of rotation, the ellipticity as well as the den- 

 sity of these shells varying from the centre to the outer surface, 

 and so being a function of the variable polar semiaxes c', then 

 it is easy to see that the potential of such a heterogeneous 

 spheroid is 



3 r lo ^ ^ ^ r' 

 where c is the polar semiaxis of the outer surface, and 



,, , r- d.c'\l + 2e) ., , , , C'^ d{c'Ke) , , 

 ^('^'^ ] P ^? '^' '' '^^'^^j P dc' "^^ • 



In the theory of the figure of the earth it is shown that the 

 condition of fluid-equilibrium implies the relation 



(-1) 



<;>W-§f(c-) = 0, 



e being the ellipticity of the external surface, and m ihQ ratio 

 of the centrifugal force at the equator to the attraction of the 

 earth there. The value of m is, neglecting small quantities of 

 the second order, given by the equation 



where o) is the angular velocity of the earth. 'Eo^Y, since 

 M=-|7r(/)(c), we get for V the form 



^j M .Ms/ m\P-2h 



This value of the potential is arrived at on the hypothesis of 

 a certain distribution of matter in the interior of the earth ; 

 but it is important to remark that it has been obtained by 

 Prof. Stokes (Cambridge and Dublin Mathematical Journal, 

 vol. iv. p. 207) without any such hypothesis as to the distri- 

 bution of density, be the interior fluid or solid, but assuming 

 only that the surface is a spheroid of equilibrium of small 

 ellipticity. 



