336 



C = ^{a' + a,^Di-l) + a2HD-D,)\ 



+ aiai3(Di-l)(^|jJf7. + gf73 + &c.)} 



C being an arbitrary constant. 



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

 and by hypothesis a - a\, a\ - a-z, 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-'^{^.^4iD.-X).'^(D-D.)).Ua\ 



and 



+ ^^^^'" y - 47r [aa^ {\Y-i + l \\ + &c.) 



+ aiOi^ (Z)i - 1) (iC/o + I Ui + &C.)J 



+ \ ga^ (cos'0 - *) = 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, U^ = 0, U\ = 0, and in general 

 Ui = 0, when i is not 2, and ai Uo = - e\ (cos'-0 - ^) ; ei repre- 

 senting the ellipticity of the spheroid. Hence 



3 u Yo 3a , , ,^ „ . 



C-^^-*i^)(-^-^)- 



But also 



^/ = y, + Yi + Fi + . . . y^. 



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

 sions, we obtain 



