PROFESSOR O. H. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
When .■!; is odd (000, EOC) the only change is that i{i + 1) eaters with the 
opposite sign, so that the first fraction ends with 
- 3} {i, 2} 
— 1 — /3<j — \l3i{i + 1)' 
When s = 1, we have + 1) equal to the same fraction as before. 
When the q’s are determined we have q'f = tqt. But it is desired that in the 
case (a) q^ should be unity, and that in the case (/3) q's shoidd be unity. This con¬ 
dition will be satisfied in the present case if we determine all the q’s, put equal to 
unity, and finally take 
, s 2n 
5 * ± 2,1 ± 2 ,' 
o 
Thus in both (a) and (^) we put q^ equal to unity, and in (/3) determine the q'’s by 
the above equation. 
(y.) We now have to consider the cosine and sine functions. 
For EEC, EES, OOC, OOS 
O’’ r cos , , ^ ^ r cos - ^ \ ^ n r ^os, , ^ \ , 
sin 1 sin ^ ^ 1 sin 
The first S has limits Ts or (s — 1) to 1, the second T {i — s) to 1. 
Apply the operation y, and equate — ’ then 
- S87i(.s — (-5 — -f SSn{s + + (i’ + 27i) 
+ Ps 
O'. .9 +1} y“qs + 2) .^ + 2 ,t .•«(. + <i .»} I™ (s - 2) ^ 
-f S/B'‘p {i, s - 2 / 14 - 1 } {s — 2n + 2) <f) + 2a- {s — 2n) <f> 
{sin 
4 - {?:, .s - 2w} {s - 2n - 2) (j) 
fi - + 2)1 
{'/, s + 2?i 4-1; 
J cos , 
[sin '' 
s -f 2w H- 2) (/) -f 2o- -j {s + 2//) (p 
cos 
-j- {/, s-f 271} {s -f 2n - 2) 4> 
.3 Q 2 
= 0. 
