FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
O 
o 
Iii this equation we may put f+<f> = 0, or, since </> = u+fv, we may put 
/ = 
1 +u’ 
although of course it would not be legitimate to equate differential coefficients of these 
equal quantities. The equation reduces to 
u 
3 a \i + 
v 
+ 2 
u 
3£U + u 
= 0, 
( 10 ) 
and this may be readily solved in powers of the parameter e on assuming a series of 
the form 
u 
l+v 
= egi + e?g 2 + e 3 SL + eY/i + 
• ( 11 ) 
Equating coefficients of the different powers of e we obtain 
= o 
Sx ’ 
3(72 _ 1^ 1 /3<7i 
3a ~ 4 ^A 2 \3£ 
3x 4-J A 2 \ 3f 3£/ 
so on. 
To satisfy the first of these equations, gi must be a function of f, £ only, say P. 
To satisfy the remaining equations, write 
A = — —i-, &c., so that ^ = -k, &c. 
a 2 A’ ’ 3a A 2 ’ 
Then if P f is written for 3P/3£ the equation for g 2 is 
3</s _ jly p 2 
d. ^ -t £ ? 
3A 
3a 
of which the solution is 
g 2 = -i(AP/+BP/+CP f 2 )+Q,. 
where Q is a function of £ >?, £ only. Proceeding in the same way, we find 
g, = * (A 2 Pf 2 P« + ... + 2BCP„P f P, t + ...)-i(AP t Q { + ...) + R. 
where Id is another function of £ > ? , £ only, and so on. 
b^2 
( 12 ) 
(IS) 
