FIGURE OP EQUILIBRIUM OF A ROTATING MASS OP LIQUID. 
2G1 
L* = 7 (1)* ff U + 22 e!/;>,s;,^.-/)] ae d,i>. 
The hrst term vanishes because /S'/ is a surface harmonic and fE> cW d(j) is proportional 
to 
Thus we are left with 
f/" = ~ 7(4Tff + itt‘fis,snded<l>- 
I now introduce symbols for certain integrals, and in order to bring all the 
definitions together I also define several others which will only occur later. 
Let 
7' = [)''t>(s/rrf»# 
0 ,,' = 
, _ cos= /3 cos» 1- II (S,YSf de d.j> 
sill /S 
( 8 ). 
All these inteuTals vanish unless i is even. For immediate use 1 also introduce 
= II T'TS'g/S'/ cW d(f). 
The xfi integrals vanish unless i is odd, but it will appear later tliat they are not 
actually required. 
I further write 
cr^ = 11 q/' (SqY dd d(f), = 1fl 
6 cus'^ /3 cos* 7 sill /S 
0-4 = 
siir 7 
^ ^ - 3G ^ 
^ 
1 
(»). 
VA/r/ AiMV; 
With tliis notation we have at once to cubes of small quantities, 
1^ p dv = — M (Jj-^ [cV^ + 22:e/-"i///].(9). 
But before using this 1 will obtain another integral to the fourth order. It is 
I rp dv = M (TT f f + 2Sey,>S,S,- + (2/‘S,‘F] 
. r \ / J J 
+ {S,f + 3Sey/ {S^S^ - cie^ {S,f} dO dcp. 
Emitting terms which vanish, amongst which are integrals of the type we 
have 
