PROFESSOR G. H. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
470 
We shall now consider those terniinatitnis. 
First, su])j)ose that .s is even, corresponding t;) types EEC, OEC. 
The lirst continned fraction depends only on the first S. The condition to l^e 
satisfied is 
+ ... 
+ /3^*(Zo[P^ - + {h 0] [l - 1}P-"] 
+ - 2cxP'^ + {h 2} [i, I] P] 
+ _ 2aP* + (C 4} .p-, 3}P-^] + ... = 0. 
Since {f, 0} {f, — 1} P “ = P-, we have, by equating’ to zero the coefficients of 
P and P^, results which may be written 
'Is 
-K21P, 1} 
T- - /3a 
li 4} {h 3} 
•s- — 4 — /3a + 
i 
Hence the q's disappear from the tirst continued fraction, which terminates with 
2} KH 
s~ — /3 a 
In this last term the ^/3- which prevails elsewhere is replaced by 
Observe that when 5 = 2 the first continued fraction is replaced by a simple 
fraction, so that the equation for ^cr becomes 
/5cr 
11 
- /S 
a 
+ i/3^ 
H- 
31 
Secondly, suppose that s is odd, corresponding to the types OOS, EOS. 
The condition to be satisfied is now 
2(.r - I)ySi(*-')( 7 iPi + 2(5- - 9)^^<*-^'h_/3p3 + ... 
+ - 2o-pi + [f, 1] {i, oj P-yl 
+ - 2c7P3 + {d 3] [i, 2] pi] 
+ - 2o-P + {/, 51 {/, 4} P3] + ... =. 0. 
Now {i, Ij- [?,, 0} P ~ 1 f (f -|- I) —— ^ P “ 1 = f (i 1) Pb aiid if we equate to 
zero the coefficients of P^ and P® we obtain results which may be written 
2'?] „ _ — {d 3} {i, 2} 
'73 S“ — 1 — /3a + ^/3i{i + 1 ) 
2 '/:^ _ — { /, 5} {/, 41 
25 .p _ 0 _ ^ ) 
