FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
5 
equation (16) will be a solution of equation (15). Moreover, if iv\ tv", w'" ... all 
vanish when A = 0, v will also vanish when X = 0, so that equation (14) will also be 
satisfied. It follows that equations (16) to (19), with (23), contain a complete 
solution of equation (8). 
6. These equations can be solved in powers of the parameter e. Let us assume for 
u, v expansions in the form 
u = eu\ + e 2 u 2 + e 3 u ;i + ebq + ... , 
v = ei\ + e 2 v 2 +e i v 3 + e i v i +... , 
and for w, w\w", ... &c., expansions 
w = ewi + e 2 w 2 + e 3 w s + e i w i + ... , 
w' = ew\ + eW 2 + 6 3 w' 3 + eW 4 +..., 
The coefficients in the expansions of u, v are of course not independent of those 
in the expansion (ll) already assumed for u/(l+v). We find easily enough the 
relations 
Ui = Qi = T, 
u 2 = g 2 + vigi, 
u-s = g s +v 1 g 2 +v 2 g 1 , 
^ = g i +Vig :i +v 2 g 2 + v 3 g 1 , &c. 
The value of 0 (equation (21)) is found to be 
0 = ^ \e 2 gi (ut\ + 2fw'\ + 2>f 2 w"\ + ... ) 
+ e 3 {g 2 {w\ + 2fw\ + Sf 2 w m 1 + ... ) +g 1 {w’ 2 + 2fw" 2 + ...)}]. 
On equating coefficients of different powers of e in equations (17)—(19), we obtai 
am 
y x_ 9 u\ 0wq\ _ 
i (0C 0 IV i CtV i p 
4 (~a aWvvU 
(24) 
(25) 
«!!?*!?)--’'--“I 
f ^(w\ + 2fw'\+ ... ) 
(26) 
1 
A I V' 2 . CIV 2 \ 1 r-72 P 
( 27 ) 
