484 
PROFESSOR G. H. DAR^YIX OX ELLIPSOIDAL HARMOXIC AXALYSIS. 
If ^ye equate to zero the coefficients of (s ffi 2n} we find 
2jys -•>»_ {h s — 2n + 2 } _ 
4^^ (.5 _ n) - I3a - 1/82 s - 2?l - 1} / -2A-2n-2 \ 
\ P.-2« / 
2ps + <>„ _ _ ■ -{i, 5 + 2 ?;, — 1 } _ 
Ps + nn-i 4 , 2 , (.5 4 . n) + /3cr 4- {i, s A 2«, + 0} 
\ ]?s-{-2n / 
These will, as before, lead to continued fractions, and by elimination of the jj’s to 
an equation for /3cr. The equation will agree with our former I’esult, for it can of 
course make no difference from which equation we determine o-.'^ It follows then by 
comi^arison with the previous result (16) that 
Vs —in _ ^ ill 
Ps-2n+i ~ id S - 2ll + 1} qs-2n + i ’ 
Ps + in 
2's + in-i 
S 
2;i 
1 ] 
+ ill 
2-s + 2)i - 2 
Hence when the </s are found, the y>’s follow at once. 
(8.) For OEC, OES, EOC, EOS 
S' 
= U) 
rj \ 2 4 -1. -I : is - ■2n)<f> + sfi-pi ,i(«+ 20 <#. 
cos 
SlU 
cos 
sin 
where <P = (1 — ^ cos 2(^)d 
The limits of the first S a^e T.5 or (s — 1) to 1, of the second — s — 1) to 1, 
Proceeding exactly as before we find 
2p's-in 
P s — in + i 
‘~>i/ 
-r ^■ + 2 « 
P s -^ii,-i 
{/, S — 2)1 + 1} 
4?? (.s’ — — /So- — 1 ^ 2 _ 2)1 
— \ i, s + 2/d 
^2p's-in-i ' ’ 
P S - in : 
4?i (-S' + ))) + /3cr + f-/3" {?, -s’ + 2ii + 1} / ^ 
\ iG+ ill 
By comparison, with (16) we see that 
P s-i,i 
P s-'ln + i 
/ 
P « + 2i) _ _ ( ; 
/ 
P s + in — 2 
f 9 s —in 
{o ^ 2)1 + 2j- 
— [i, s + 2n 
■> 9s + in 
*^')l __ 
+ -2 
Therefore when the <i'& are found, the 7 /s follow at mice. 
* 1 have cf course verificl tliat this is so. 
