1883.] On the Figure of Equilibrium of a Planet, tyc. 159 



Let h be the ratio of tlie density of the stratum a to the mean 

 density of all the matter situated inside that stratum, and k the 

 surface value of 



Let C, A be the greatest and least principal moments of inertia of 

 the planet about axes through its centre of inertia. 



Let ce be the ellipticity which the surface would have if the planet 

 were homogeneous with density p, so that ce—^m. 



The condition that the surface of the planet is a level surface is 

 satisfied by — 



C-A=§Ma 2 (e-±m) (1). 



The condition that the internal surfaces are also surfaces of equi- 

 librium demands that e should satisfy Clairaut's equation — 



(^-6—\[\vahla + 2iuc^(—+^)=6 .... (2). 

 \da~ a 2 / J o \da a) 



It may be proved from (2), and the consideration that w must 

 diminish as a increases, that e cannot have a maximum or minimum 

 value. 



Also it may be shown that the constants introduced in the integral 

 of this equation must be such that — 



ce 5m 1 d , 9 n /0 n 

 - = — = (ea~) (3). 



when a is put equal to a after differentiation. 

 The mean density is given by — 



a 3 /3 



=3j" wo? da (4). 



And 



P 



Neglecting the ellipticity of the strata, we have the moment of 

 inertia about any diameter of the planet given by — 



c=f.j 



a 



iva^da (5). 



The ratio of (1) to (5) gives the precessional constant. The pres- 

 sure and density are connected by the equation — 



l(-^ + ^7rwa)da + ^-[ a wa^da=0 .... (6). 

 J a\w da J a Jo 



* Jc, fc, are the reciprocals of/, f, according to the notation adopted in Thomson 

 and Tait's "Nat. Phil." (edit, of 1883), § 824. 



