FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
‘267 
7. The Energy ^CC; Result for \JJ — JR + CR — ^CC. 
From the last section it appears that the potential of C at the surface, where 
T = 0 , is 
u = - 3 (^3' + 2 {e- ^ + f/j a.'S’ 
For the mass of an element of the surface density we have 
pfdor 
m 
\ j * j 5 I a -;; 1 —y- ,, 
V^O/ I \ yi 
\ ^*^3 + 3 + S 3 + /f) r 
These are to be multiplied together and half the product is to be integrated. 
Then bearing in mind that |j ^ dO d(f) = 3, we have 
iCC=d{ fj + S (e^ ^7 + /‘) 
Ill the terms of the fourth order we put {k/kuY equal to unity; thus 
ioc = if(+ ff {«* i 3[„ + sa.*^ + +s (M'S.-h [. 
^’o 
/^■ol L 
(coff 
l>! J 
Combining this with (17) 
CR -\CC=i f ( Y) e%.^3 + i f P‘ 
^0 \^’o/ 
(^ 3 )^ + ^ + 13/p/) ^3 
+ te^f:\ 2 {%! + 2 ^ 3 ) 01 / + (15/ + m,)p!] + t{fdY^i<i>n . (is) 
We are in a position to collect together all the results obtained up to this point. 
Now \ JJ — JR, as given in (15), contains Pi^Q/, latter of these is what 
is now written and since the ellipsoid is critical Pi^Q/ = = '^ 3 - 
Collecting terms we find that the terms of the second order disappear, and that 
yj - JR + CR - i(7C= f 
/I'l 
+ te/c [ 2 a>; + (B/ + 233 ,)p-] - s {f-f (% - g.') <;■/} . ( 19 ). 
% (i (o-.)® + 2£*) + s (g>‘ + ».>■) Pj 
The reader will recognise that the last term involves the coefficient of stability for 
the deformation *S/. It is important to note that if Si is of odd order there is no 
term with coefficient efj 
2 M 2 
