OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
319 
energy of the layer of deformation ; secondly that due to the ellipsoid and the 
layer ; thirdly that due to the change in the moment of inertia. 
If a subscript I denotes integration throughout the space occupied by the layer, 
U the potential of the ellijjsoid, and dv an element of volume, 
iE = + [ Updv + la.- f (if + #) pdv. 
h -'I U 
If I denotes the thickness of the layer standing on the element da-, the first 
of these terms is | - . 
The value of U throughout the layer is equal to Vq — g^', where 
Vq is the constant value of U \ (jr {if + 2 ^) over the surface of the ellipsoid, and 
is the distance measured along the normal to the element d^da- of volume. 
Hence 
\updv+ io."- \(f + * 2 ) pdv = f_f p (U - gC) dCdo- 
Since Vq is constant and the total mass of the layer is zero, this is equal to 
- ip 
It follows that 
1 _ f Yij 
^p)^^ da. 
BE = Ip^ 
The axes of the ellipsoid have been chosen so as to make our original E 
stationary, and the further condition to be satisfied is that BE shall he stationary. 
Let us suppose that 
i = { 9 ) C' 
which expression shall Ije deemed to include any one of the other types of harmonic. 
Then it is shown in (51) of “Harmonics” tliat the potential of this layer at the 
surface of the elli|)Soid is 
y e l$f (pq) ( 1 ^ 0 ) Wf {9) ((^). 
Since the mass of an element is j;ep (p.) Ct {<f) da, we have 
1 -2 r _ 3 2 
k 
With the value of g found in (36) 
Mp 
13/ (>'o) Q/ {>'(.) I [13/ {g) ®/ 
Ip f gfdiT = i y P,> (v„) Q,> (a„) I 92 (p) e/ pdcr. 
