FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
2G5 
It is now necessary to express S/po in surface harmonics. The first two terms are 
already in the required form; for the remainder let 
cos-Z^cos^y /1 1 
^ 2p 2 2 "1~ p 3 ) v^3/ • 
Multiplying both sides by dd d(f) and integrating, we have 
>,,■ 0 / = eos‘=/3cos=y + jt, - (?)(,$',)= A/, 
Therefore rji' = and vanishes unless i is even. 
AVhen f = 0, rjQ = ^; and since by (7) "o = || {S^Yddd(j) = o-j, 
Ave have tjq = ^o-g 
Hence we have 
S = - Pop 
+y:-i^ 
This is expressed in surface harmonics, the middle term l)eing of order zero. 
By (5l) of “ Harmonics” the internal potential of 8 is 
'i, (0 ©3 (-o) 'S's + icV.f + 2^1“ +/‘) P.' H &■ M'% 
We have W(.) = ^ ^. 1 * gut I^efore 
' ^ ^ 2v^^ cIvq ^ sin/3 ^T-Tt “L, 
proceeding to use tliis I Avill introduce a new abridgment, and let 
B7 Of 
dVn 
(16). 
Then 
V, = - 3 
M / k y 
Icq \ Icq } 
W (-) a/ (^o) = ^t-r ^, and 
+ 3 
/ k Y cos- /3 cos” 7 
7oV/. 
'^0 / 
sin /3 1 d- - Y 4>f ^ A,T\2 j 
In order to find the energy CR we multiply by the element of mass 
p dv \ M — 2 t''F] dr d9 d(f), 
\ ^0 ! 
and integrate throughout h 
VOL, CC.—A. 
2 M 
