336 



+ «iai3(Z),-l)(|^^C7, + ^^t73 + &c.)} 



C being an arbitrary constant. 



But r, the radius of the surface of the fluid, =a (1 + ay), 

 and by hypothesis a - ai, ai - 0-2, are small quantities ; hence, 

 if r be developed, and all small quantities of the second order 

 be neglected, we shall have, remembering that C is arbitrary, 



C-*-^{^*'^(D.-^)-'iiD-D.)yUa^ 



and 



^47r^a2a^_^^ {aa2(iy2 + fY3 + &c.) 



+ aiai2 (Z)i - 1) {lU2 + }Uz + kc.)\ 

 + I go' (cos'9 - i) = 0. 



" By a process exactly similar to that performed in the work 

 referred to, and remembering the assumption of the theory, I 

 find for the solid spheroid, U3 = 0, t/4 = 0, and in general 

 Ui = 0, when i is not 2, and ai t/2 = - ^i (cos^O - i) j ^i repre- 

 senting the ellipticity of the spheroid. Hence 



3aY2 3a ,, ,^ p , 

 a^=-^^ + -^(fl3 + &c.) 



-f-^^-*'iA))(-'^-*)- 



But also 



y=Y,+ Y3+Yi + ...Yi. 



Hence, comparing terms of the same order in these expres- 

 sions, we obtain 



