FIGURE AND STABILITY OF A LIQUID SATELLITE. 
191 
Hence the contribution to (ISj) corresponding to this term is 
3 
k 
X 
(l+X) 
Now P/(1 )(£/(t7t) vanishes for all harmonics except cosine-harmonics of even rank. 
Therefore 
X 
(i +M ! 
C/ / I x f i even 
cr 27r) Uodd • 
(18), 
the summation being for all values of i greater than 2, and for all even values of s. 
The lost energy (Lsi) is expressible by the similar function for the other ellipsoid, 
but I have not adopted a specific notation for the ellipsoidal harmonics for the 
ellipsoid E, and therefore cannot write down the result. 
The lost energy (e + S 2 + ^a)~(if + z 2 ))l is the potential of the ellipsoid e, together 
with the potential of S in as far as it involves harmonics of the second order, and 
a rotation potential, multiplied by the density of the layer l, and integrated over 
the surface of e. That is to say it is the potential of gravity on the ellipsoid e 
integrated throughout the layer l, which it is not permissible to regard as surface 
density. 
If the thickness of the layer be £, and if cl £ be a slice of that part of the layer 
which is erected normally on an element dcr of the surface e, then p da is an 
element of mass of the layer. The potential of gravity is —gC Hence the lost 
energy is 
- P \\gtdtd<r = —\p j* gC da. 
Accordingly the lost energy is equal and opposite to the work done in raising the 
layer, considered as surface density, through half its thickness, against gravity. 
We may take as a typical term 
l =pMi'(p)W (<!>)> 
and we have shown in (13) that 
3 4 3 X 
9 = Jk VTx 
A 11 y cos!i r 
1 3Xr 3 cos'/3 _ 
1 + 
3(3 +;c 2 ) F 5(5 + 2F + F) F 
14/c 2 r 2 
56ft 4 
It should be noted that this expression for gravity takes no account of the change 
in the ellipticity of e which is due to the fact that E is an ellipsoid and not a sphere. 
The error introduced thus is however outside the limits of accuracy which have been 
adopted. 
Accordingly this portion of the lost energy is 
- ^ IV a ' 3 (//) 2 { A, 1 + series} | [f/(/^)Cf (<£)] 2 p da. 
