1882.] Liquid Ellipsoid about an Axis. 215 



Also B= toP°te f*«n'_fp, 



= i^ f 1 _ _ ^>cosj> 



(Legendre, Functions Elliptiques, I. p. 257) 

 47rpV{a?y(l-y)| _ iirp Jw/ 4-rrpy 



and = 4 ^ a6c ^tan 2 ^ 



= , , >Nfl pa (tan </> A</> - £</>) 

 . (a 2 - c 2 )- K 



±irp Jxy [ V(l - ?/) \/.g _ „, | 



_ 4f7rpx 4irp Jxy _ 



_ a-y (x-y)J(l-y) 



Therefore 



A _ V^ y (1 - y)} gr. V(^) 



E<f>; 



4-rrp (\-x){x-y) r (l-aj)V(l-y) r a-y 



4?rp x-y {x-y)s/(l-y) 



and A + B+ G = 4irp. 



Therefore, in region I., 



x — A 2 (/>, y = cos 2 (f>, 



A _ cos $ A0 i^) cos A ^ A 1 ^ 

 47rp sin'^ sin 3 <£ sin 2 #sin 3 </> 



5 cos </> A</> £"0 cos $ A0 7' 7 cos 2 



47rp sm 2 #cos 2 #sin 3 (£ sin 2 sin 3 <£ C08 8 #sin 8 d> ' 



C^ A 2 <fr _ cos A0 #0 



47rp cos 2 ^ sin 2 cos 2 #sin 3 c/> 



