548 
PROFESSOR O. H. DARWIN ON ELTJPSOIDAL HARMONIC ANALYSIS. 
It may l)e well to remind the reader that > 
i (i + 1) 
% = 
r= 
s~-l ’ 
(i — l)i (i + 1) (i+2) 
.s2-4 
./ = ^ (^‘+1)- 
First wlien s > 2. 
f U'A)] 
^ttM i + .s! 
'2/+! i—R 
H ; 1 -i/3'+-,',;;e-[22H3S-s-(2S-1 )+T] 
Tyi'Ps |p(?P,'W I 
Pt^"{ees 
+. 
2’pM y«_: ji_ps+-,'„-^=[2S-+S-C-r(25;-l)+T]j, 
2t+1 
d(T 
^ {' + P(2 + 2)+-i‘ff/3-[l n +10 + *5(25-- 2S + I)-T]< 
I "i % St 
C, 
S; 
(/cr 
(l+l^(5;+2)+-,'s,8’[9S + 4+s':(2r--2'+l)-Tj} 
? *T“ -L % *“ S » 
Secondly, M’hen s= 2 , S = (^-f- 1 ). 
'PvpeEEC |p(^p?(p)®.=(,^))'W=|’;,-‘f;^;{l-i^i + rfir/3-(05S:-n82 + 
r SttAT 7* -4- 2 T 
Type OEC j p (^0/(,,)Cy(.^))%?<r=^y_- ■' 1 - i/32+5iir/3'(93S=-1783; - 40)-;. 
Ty|)e EES Jp(P'(p)S“(^))'Wo- 
4f+T h ^ 7b/8-( 292H 9 OS + 210! 
Type OES Jjj(P'( p) Sf^fda- 
27rM i+2! 
2i+l i-2! 
; {l+P(S + :^)+2io/3'(29N^+106S+12G)}. 
Thii'dly, when .?=1, S is infinite and we must nse/=/(^-f 1). 
Type OOS f p (jp '(p) S,‘(4))'Vo-=|^3 ) 1 +i^ (./-2)++T/?’(./'-2fi/+-19)!. 
