1879.] liquid ellipsoid about its mean axis. 241 



The shape of the surfaces of equal pressure therefore changes 

 every instant, unless we can make 



(c'-a'){Sc' + a') , {a'-¥){a' + Sb') , 

 (c' + ay ""^ {a'+hy '"^' 



and the two similar expressions constant. 



This can only be the case when the ellipsoid is rotating about 

 a principal axis, say the axis of z ; then oo^ = 0, (o^ = 0, and &>3 = « 

 suppose, and we can make the surfaces of equal pressure similar to 

 the external surface by putting 



^^^- — (^qr^y — '^l 



= h'\B + 



{a'-h'){Sa-'+b') 



(O 



or 



(a'+by 

 a'A-c'G c'C-¥B 



^ {a'-¥)ia' + Sb') , {a'-U'){3a'+b') 

 ^ (a' + by {a' + by 



Hence c must be the mean axis, and we can therefore have an 

 ellipsoid of liquid rotating in equilibrium about the mean axis, the 

 motion of the liquid particles being that which would be generated 

 in the liquid from rest. 



Therefore 



a'A-c'G c'O-b'B 



a'{a' + Sb') ¥{Sa'+¥)' 

 or 



aV (Sa'^ + b')A + a'¥ {a' + Sb') B - (a* + Qa'¥ + b') c'G=0, 



"a^HSa^ + &2) (J2 + X) (c2 + \) + a^b^ {a^ + 352) (^2 + x) (a^ + X) - (a^ + ea%^ + b^)c^a^+\){b^ + \) ,. „ 

 — dA = U, 



where P' = (a' + X) {b' + X) (c' + X), 



of which the numerator N ' 



= X' {4<a'b' (a' + b') - c' {a' + 6a'b' + b')] 



+ \ [a'¥ (3a' + b') {¥ + c') + a'b' (a' + d¥) (c' + a') 



-c'{a' + b'){a' + 6a'¥ + byi. 



