190 
SIR G. H. DARWIN ON THE 
+=r 
du 
(u + a 2 ) 12 (u + b 2 ) 112 (u + c 2 ) 1/: 
A Q A. 
e — f 77-pa 
§ 8. Remaining Terms in the Expression for the Lost Energy. 
If e be the mass of an ellipsoid of semi-axes a, b, c, the lost energy of its 
concentration is 
i i (ee) = 
where 
In the present case 
and 
where 
Thus 
By symmetry 
where 
and 
+ X’ 
, 2 r dv 
* k J„ 0 (v 2 —l) 112 (v 2 —1/k 2 ) 112 ’ 
1 1 
V, = -7- = ---= . 
k sin y sin p 
2 ( ee ) io ) ( 1 + x) 2 ^ • 
... (16). 
2 {EE) 1 3 0 ( 373 -pa 3 ) 2 ^ ^2 'T . . . . 
i—1 
2 P dN 
^'n„(N 2 -1) 1/2 (N 2 -1/K 2 ) 1/2 ’ 
N„ = - 
1 
1 
K sin T sin B ' 
The lost energy (IS X ) is the potential of a particle S, equal in mass to E placed at 
the centre of E, with the omission of terms of the second order of harmonics, 
multiplied by the density of the layer l and integrated over the surface of e. This is 
the same as the potential of the layer l, with the omission of harmonic terms of the 
second order (and there are none such) at the centre of E multiplied by the mass 
of E. 
A typical term in the surface density representing the layer l is, say, 
fm (ri&f it)- 
The external potential corresponding to such a term, at the point v, y, cf>, is by (51) 
of “ Harmonics,” 
The co-ordinates ol the centre ol E are v = 1 -^,y = l,(j> — ^77 ; and the mass of E is 
l 7 r d a 7(l+X). 
