186 
SIR G. H. DARWIN ON THE 
terms in the potential of S at the surface of e excepting terms expressible by 
ellipsoidal harmonics of the second order with respect to the ellipsoid e ; and S 2 , which 
is to contain the omitted terms of the second order. Similarly, the term sL is to be 
divided into s x L and s 2 L. 
Take the centre of e as origin of co-ordinates x, y , z with the z axis passing through 
the centre of E, the y axis coincident with the mean axis of e and the x axis coincident 
with the least axis of e. 
Since l is expressible by harmonics higher than the second order, and since y 2 +z 2 
is expressible by harmonics of orders 0 and 2, it follows that the moment of inertia of 
the layer l about the axis is zero. If therefore oj is the angular velocity of the system, 
a contribution to the lost energy of the system which may be written symbolically 
[^co 2 (?/ + z 2 )] l is zero. 
It follows therefore that we may write 
(e + S)l = [e+S 2 +^co 2 (^+z 2 )]l+S 1 l. 
Similarly, if the ellipsoid E be referred to a parallel co-ordinate system X, Y, Z 
through its centre, and such that 
x = X, y = Y, z = Z+r, 
so that r is the distance between the two origins, we have 
(E+s)L = [E+s 2 +±a> 2 (Y 2 +Z 2 )]L+ Sl L. 
The problem is already so complicated that it will be convenient to omit certain 
small terms in the expression for the lost energy, which it would be very troublesome 
to evaluate. 
The term (e—s) L represents the mutual energy of the departure from sphericity 
of e with the layer of surface density L on E. This term is clearly very small and 
will be omitted. Similarly (E—S) l will be neglected. It will appear from the 
results below that these terms are at least of the seventh order in powers of 1 Jr. 
A fortiori IL, which is at least of the eighth order, will be omitted. 
The whole expression for V will now be divided into several portions. 
Let (eE) 1 be that portion of eE in which each ellipsoid may be replaced by a 
particle; it is, in fact, the jn’oduct of the masses of e and E divided by r. 
Let (eE) 2 be the rest of eE. 
Let (vv) denote that portion of V in which the larger body E may be replaced by a 
sphere; then 
(vv) = %ee + ^ll + [e+S 2 + %(o 2 (y 2 +z 2 )~]l+S 1 l. 
Similarly, let 
(VV) = ±EE+±LL+[E+s 2 +±a> 2 (Y 2 +Z 2 )]L+s 1 L. 
Then V= (eE) 1 + (eE) 2 + (vv) + ( VV)+ neglected terms. 
