rROFESSOR G. H. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
537 
Thus the second term of D is 
(-) 
+(-)*"' 
” + i 2n ! 
‘ 2 ) 1 1 
F ^^+1, 
‘2~’>(n !> 
[(2».+ 1)...C^^^ + 2-s-1)T 
'2-^()i +1).. .()i +s).5! 
H + S ^ ^ 1 
4 X 27Z7 ~ ^ ^ 
L ii + W^ ,< + l * 
K' 
The first term of this becomes infinite when € = 0, but it is equal and opposite to 
the infinite term in the first part of D. and they annihilate one another. 
Hence 
1) = 
n I n ■ 
l!'^ h-y[^-3...(2r-l) P 
2/0 
/y, 2/1! 2,t y [( 2/< + l)...(2// + 2.b- —1)]- 
4 ) 2-Gh + I). 
4 ',1+ . S , ,! + ..> , - 
“>og;+^7+^r-4^^ 
K' 
‘(v l)- ' 2-^(h +1)...(??+0. .s! 
On examining the case of n = 0 we find that this formula also emV/races it, 
0 
]Dro^dded M^e interpret X meaning zero. 
1 
The coefficient in the last term admits of some simplification, for 
/l + 
1 ^ 1 ■'4+'’ I 
^7+W-4^2/W = - 
1 1 1 
" :' + ,<r 2 
t=i} 
2T - 4-X - 
AVe thus coiiclude that 1 > 
or 
co,s""^ 2-“)il n-i: 
- <W— --- 
:a 
D. 3^ 
--_ 
2H))-^)\\ ^ 2h)?-])(/i-2)2: 
... to n terms 
+ (~) 0- 
■' -L/l 
r 
2~»{))\y 
1 
+ 1 
(2h + 1T .4 1 „ 
■2Hn+ l)liv 
(2/i + lfi(2/. + 3y / 4 '^I 1 
■^2‘e/? + I)(>? + 2)2:^" ^/c ^ t t{2t-\) 
3 . 1 
)k^ + 
. (84). 
'fhe second integral E may be found as follows :— 
E„= fcos2" 0 Af/^ = J‘cos2" 0 [k2+(I~k=) cos2 0] 
— K-D;;d-(I — /c') . . . . 
YOL. CXCYI].—A. 3 Z 
(85). 
