. 188 
SIR G. H. DARWIN ON THE 
But | is the potential of the ellipsoid E at the centre of the ellipsoid e, and by 
an exactly parallel transformation 
f da -ES ( - y ‘ 3 E 2n ( 1 82 i ^Y 1 
J It - t*" , Q\0„1 W \ 
0 (2?i+l)(2n+3)2n! V \K' J dx 2 + cy 2 J p 
where p 3 = (x-Xf+ (y- Y) 2 + (z-Z) 2 . 
Since our co-ordinate axes have a perfectly arbitrary origin, we may at once put 
X = 0, 1 = 0, Z = r, z = 0, and after effecting the several differentiations put x — 0, 
y = o. 
It follows that, on putting x = 0, y = 0, after differentiation and writing 
2 2 . 2.2 
p — x + y + 1 ', 
_ (-)* 3 _ ^(i Z+ZYi (-)* 3 wLZ . ilYI 
o(2n+l)(2ri+3)2n! V 0a; 2 dy 2 J f (2i+ l)(2i+3)2i \^ U 2 0a; 2 0y7 p‘ 
(mE) = eEl 
Y 02 02 -j 02 02 
If we denote the operator + by d 2 , and the operator —-—■+X- by D 2 
tr dx 2 cy~ J 1 K 2 dx 2 dy 2 J 
we have 
(eE) = eE 
= eE 1 - W+ 
- 2^7 (W + A“D 6 ) - —dy (* ‘+i‘^d‘D“)... 
On effecting the several differentiations, and putting x — 0, y = 6, we find 
d*l= + d‘I = l/^ + 2 \ D s d 1 i = -,( 3 1 _1_ + 3 \ 
P r s W 1 p r 5 \« 4 K“ / p r 5 \k K 2 K 2 K 2 / 
d-I = _5!p/5 8 8 V d . D U = -^/5 12 2 
p r 1 \k k k J p r‘ \k K" k k“K“ k“ K" / 
and the remaining functions may be found by appropriate changes of small and large 
letters. 
If now we again use p to denote the density of the spheroids, and revert to the 
notation employed elsewhere, namely, 
— _ T? — 1 
• ~ 3 
14 X 
1 4 X ’ 
