180 
SIR G. H. DARWIN ON THE 
The higher harmonics will be considered later, and for the present it is only 
necessary to consider the terms defined by i = 1, s = 0 and i = 2, s = 0 and 2. 
The following are the definitions of the several functions, in accordance with 
“ Integrals ” :— 
PoM=i, PoM = i, = 
tyl (v) = V, % (p) = /X, Cl ((f)) = y(l-K /2 COS 2 (/)), 
ip/( o = ^-' 4 . a*w = ®/ w = cos 2 ^ («= o, 2 ), 
K. 
where p s 2 = ^-[1 +k 2 + D] and D 2 = 1—/cV 2 , with upper sign for s = 0, and lower 
for 5 = 2. 
Hbiicc 
»i(l)ffii(M = 1. ®/(l)®/(W = (* 2 -g,') 2 .'* (* = 0,2). 
Then from “ Integrals,” equations (5) and (6), 
®i = l, E 2 s = |J[i> t ±(l+|«V 2 )(l-2K /2 )i>] (5 = 0,2). 
O 
Thus as far as the second order of harmonics the potential of M/X at v 0 , p, <j> is 
M 
a j ^0 (j:) +3©! (L j % ( Vo ) % (p) Cx (<£) + ± q 0 '»® 2 (L\ („ o) % (/i) c 2 
+ ^ (« 2 - qi) W (y ) W (v 0 ) (p) G, 3 (<£) 
We must now express the several solid harmonics involved in this expression in 
terms of x, y, z co-ordinates of a point on the surface of the ellipsoid. 
We have 
% M % it 1 ) Ci (<i>) = cos 2 <f>) = |. 
By the definition of ellipsoidal co-ordinates the three values of or which satisfy the 
equation 
+ ^ — k 2 = 0 are fi 2 , p 2 , -i (1 — k' 2 cos 2 (/>). 
K 
O) 2 — 1/V ofi- 1 OJ 
Hence we have the following identity 
X * ,_ lf_ _, ^ _ 7,2 _ 7.2 (^Q 2 — W 2 ) (p 2 — 03 2 ) ( 1 — K /2 COS 2 (f) — 0)V) 
W"— 1/fC 2 03“—1 03“ (oj 2 —l//C 2 ) (ct3 2 —1) 0> 2 /C 2 
Putting oj 2 = (5 = 0, 2) we find 
f / w W (p) c 2 s (</>) = tf ,v 
1 X 
, v .2 
2 1 2-2 
y , JL. 
2 7 2' 9 72 Q 
ql 2 k 2 K 2 —q 2 k 2 q 2 k 2 K~_ 
(s = 0 , 2 ). 
1 
