FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
283 
By the variations of and e®, we have 
= 0 . f = 0, 
^0 + ^ 
(Bff 
4 
c=> + A/, + i?/// + ^i) = o 
But from the first two of these equations 
a 
7? r ^^ 2 )^ « I 
A /3 = / ' : 
(B,^f „ ga)2 BJ^'a 
Therefore 
n z fz — (>% _ ' 
Tttp a ’ ^3 73 - ^^3 ^ c. 
‘^0 + 
U/ 
■^ 47 rp\““^ c, c:-‘^ 
= 0 
(30). 
When 8oj^ has been found, andy® are determined from 
_ -^2 3 j_ Jh 
■“ C; 47rp a 
3 ^ 
(31). 
47r/3 Cy J 
A consideration of these formulm shows that it is iimnateiial what definition is 
adojjted for any one of the harmonics, provided, of course, that the same definition is 
maintained throughout. 
In order to evaluate Aq, we must eliminate ©/. 
Since a/ = P/€l/ , Bf = mt = and 
(Ivg dvf, 
sin^ y8 
we see that 
Hence 
Bf= Bt - 
cIvq cos /S cos 7 
sin" j3 \ 23;* 
cos /3 cos 'yj 
a.- + W y + i —y- = Ar. + iKptf - i - Si • (32). 
<f),* (/)/ ^ ^i* .^/0;* - / / ^ COS /3 COS 7 0;* 
If for brevity we denote this last expression by \i, 6‘], we have 
= ^3 [x + 2 ^ 4 .] — + % [t, s] . . . 
Bt = (^>/ + iB/p/) A-[B,- 
sin^ /3 \ ., }> .... (32). 
2 COS /3 cos 7 
Ci = . 
We have now the complete analytical expressions necessary for the solution of the 
problem. 
2 o 2 
