182 
SIR G. H. DARWIN ON THE 
r \ z 
/y»* 
I Sl ^ 
■T4 ” 
( 2 K 2 -i)®^. s - ((a2+l a 2!) 
i,/' & 4 k 2 
+ f pj -(2-« ! )®i=^l_(® 1 +© il >) 
+ »£ 
4 /c 2 
( 1 + k 2 ) ^ 2 ^ + 2(® 2 + ® 2 2 )] } . . 
_ i 
(6). 
It may be observed that this satisfies Laplace’s equation, as it should do. 
It remains to obtain approximate expressions for the ££’s. 
The expression for these functions is given by 
dv 
©/ M = W 
v HPi 
'MJV-+V-1/+ 
. 2 \l /2 ’ 
We require these when v, which will be put equal to r/k, is large; thus we must 
develop in powers of l/v. 
Now 
1 
(i' 2 —1)~ 1/2 (v 2 — 1 / k 2 )~ 112 = 
2k V ' 2W 
Since p 0 (^) = 1, we have by integration 
L-\ k 
1 i 1 + K ~ | 3 + 2k 2 +3k 4 p 5 + 3 k 2 + 3k 4 + 5k 6 
^ • O o I ITo ] : I “—^—~- ”1“ , 
2 4 kV 
®o 
vW r 
i.i I + /c 2 At 3 + 2 k + 3k 1 k l 5 + 3k 2 +3k 4 +5k 6 A: 6 
6k 2 ' r 2 40 k 4 * ? A ] 12k 6 ' P + * ' ‘ 
• (7). 
There is no immediate need for this term, since it has been omitted above, but it will 
occur again hereafter. 
Since ^i(v) = v, we have 
,' r \ p 
3®, Ij =3 
\kj r 
\ 3(1 + k 2 ) k 2 3(3+ 2k 2 + 3k 4 ) A 
10k 2 r 2 56 k 4 
A 
+ . . 
( 8 ). 
Lastly, since ^/(^) = v 2 —q s 2 /i< 2 , we have 
[WW] _2 = -J 1 + % 2 + ?2fj+ . . . 
so that 
(s = 0, 2), 
[1P/(^)] 2 (^-i) 1/2 (^ 2 -i/k 2 ) 1/2 
1 + ( 1+k 2 + )- + ( 3 + 2k 2 + 3k‘ (1 + K-) y, 2 3yd \ J_ 
\ 2k 2 k 2 Jp 2 \ 8k 4 k 4 + k 4 / P + 
II we integrate this, multiply it by |L S ( ?4 ) ail( l write r/k for v, we find, 
] + (1+k 2 ) . 3y, 2 1 k? | 3 + 2k 2 +3k 4 5y s 2 yd_] L 
Sffi/fiWp 
Jk r 3 
14k 2 ' 7k 2 J r 2 
72k 4 
126k 4 21k 4 Jr 4 " 
(* = 2 ). 
(s = 0, 2) (9). 
