480 
PKOFESSOIi G. H. DARWIX OX ELLIPSOIDAL HARMOXIC AXALYSIS. 
Thus the '/s again disappear, and the first continued fraction ends u’itli 
{/, 2 } 
s' — 1 — /3cr + {i + I) ' 
(Observe that when s — 3, the continued fraction reduces to a simple fraction, and 
the equation for yScr Irecomes 
0/3Mo 3}{/, 2} 
|8cr = 
8 — /3cr + ^I3i{i + 1) 
The case of s = 1 must Ije considered separately, 
+ 5/3- [i, 5] [i, 4!- 
2:3, 
We liave next to consider the termination of the second fraction, which depends 
only on the second 
First, when i and s are either both even or both odd, the types are EEC and OOS, 
and the limits are ^ {i — s) to 1. The condition to be satisfied is 
- 2(F - - 2[('i - 2f - - 
+/3^8-%[P' + ~ - 2(7?'+ 
+ _ 2 (tP'-~ + {{,i- 2]{i,i - 3]P'-'] 
_|_ [P*-2 _ 2aP'“^ fi- {i,i — 4}{qf — 5}P'“^] + ... 
= 0. 
Now P'+ 2 jg zero, and equating the coefficients of P' and P' 
results whicli may be written 
to zero we obtaiii 
2y; 
‘Ji-Z 
- 4- + /3(t ’ 
-2;-2 
'li-i 
qi _ oy _ y + - f/3n(, t}{h i - 1}(;- 
'Mi 
h-i 
Hence this continued fraction ends with 
- - 1 } 
r - .y + /9o- 
Secondly, when i and s differ as to evenness or oddness, the types are OEC and 
Et)S, and the limits are T (f—.s'—I) to I. The same investigation applies again when 
I is cluinged into f — 1. 
Hence the continued fraction ends witii 
2} 
(i - ly - y- + /3 (t 
Tlie cases of .s = 0, .s = 1 must he considered by themselves. 
