PROFESSOR G. H. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
485 
We may now summarise our results, as follows :— 
In the general case where s is neither 0 nor I, ^cr is the root which nearly vanishes 
of the equation 
{i, s — 2} {i, s — o[ 
4.2(s - 2) - — ... 
4 /3~ {i, s + 3} I'L s -}- 4|- 
4.2(.s + 2) 4- /3a- — ... 
The continued fractions terminate variously for the various types of function, The 
end of the first continued fraction is as follows : — 
_ - 1/3- \i, s} [i, s - 1} 
^ — 4.1(s - 1) - /3(7 - 
i/3Hi,s 4- UKg + 2} 
+ 4.1(5 4- 1) + /Scr - 
For EEC —; aiid when s '2 this is the whole fraction. 
S“ — pa 
For EES - ^ ; and when 5 — 2 the fraction disappears. 
s~ — 4 — pa 
For OOC - ;——— 7 T ] and when .5 = 3 this is the whole fraction. 
For OOS - 
For (3EC 
For OES 
•s'- — 1 — /3a — ^/3i{l 4 - 1) 
-i/3m2\{i,2} 
For EOC - 
For EOS - 
5' — I — I3a 4- i/3f (i 4- 1) 
^ 2} 
5- - ^a 
^ - I/3M/,3}{/,41 
a- — 4: — I3a 
~ f/30L2Mf, 
6- — 1 — /3a — ^/3'i {i 4- 1) 
- 2}{;,3} 
; and Avhen s ~ 3 this is the whole fraction. 
; and when 5 = 2 this is the whole fraction. 
; and when 5 = 2 the fraction disappears. 
; and when s = 3 this is the whole fraction. 
; and when s = 3 this is the whole fraction. 
S' — I — ^a 4" ^/3i {!■ P 1) 
For the first four of these types, viz., EEC, EES, OOC-, OOS, the second continued 
fraction ends with 
— 1/3- { i, i} {■/, 1—1 } 
i~ — -S'- + /3a- 
; and when s — i this is the whole fraction, hut with the 
sign changed. 
For the last four, viz., OEC, OES, EOC, EOS, it ends wltli 
— ^ ^ ; and when s = i — 1 this is the whole fraction, hut 
(i — 1)“ — S' + /3a 
with the slo-n chansred. 
