of homogeneous Ellipsoids . 359 
consequence of the equation of the solid, R is a function of the 
angles p and q : therefore, making p only variable, we shall 
have 
-*y= Id) sin.^> + R cos. jcos. q. dp\ 
then, because y must be constant when % varies, we must 
make 
— dz — { sin. p + R cos. p } sin. q.dp -f { sin. q 
+ R cos. q | sin. p . dq, 
o={(£) sin. p + R cos.^j cos. q.dp { (~) cos. q - 
R sin. <?} sin.^> . dq. 
and by exterminating dp, we get 
-dz = : 
cos. q 
and hence, by substitution, 
r(«) 
A (l) = 2 ff\ cos .p sin. *p -{- cos. *p sm.p j dp . dq; 
the fluent to be taken from p = o, to p = and from q = o , 
q = s-sr. 
The transformed formula for cannot be integrated, un- 
less we substitute, in place of R, the function of the angles p 
and q, that is equal to it. Now, x = R cos.p,y = b — R sin.^> 
cos. q, z = c — R sm.p sin. q : let these values be substituted 
in the equation of the solid. 
k* k' z “r k !' 1 1 » 
and, for the sake of simplicity, let 
