500 
PROFESSOR G. H. DARWIX OX ELLIPSOIDAL IIARMOXIC AXALYSIS 
This is eas ly reducible to. 
d 
d.v 
1 
1 - ysy_ 
= 0 , 
whence 
ijtiV-’ 
1 - 
where is a constant. 
Hence 
The general sohdhni of the diherential ecjuation must 1)6 
■ (M). 
and we have already found Ijotli and Hence the two must he different 
expressions for the same thing, for the form of as a series negatives tlie liypothesis 
that it involves in the form yi^LT + 
Having then two forms of or of Q’, it remains to evaluate tlie coefficients (frf 
Ef, which are involved in the equations (40). In order to do this it will suffice to 
consider the case where v is very great, so that 
Q'-(-y 
2^ ? * 
+ 1 ! 
i + t\ 
As far as concerns the first term in the series 
w = 
211 
2d I i — si 
1 + ^'Is-2 
— S : 
i — s + 
, + /3'A -7 
'I — S ; 
- s 
I — S 
+ • 
7 — .5 ; 
= {-y~ 
2‘.il i + si 
+ 1 ! 
1 + / 3 (/, _ o .• ^; h /Sq. 
7 — .S + -I 1 
•i + S + :^! 
+ ^'ds + i- 
^ — S ; 
i - .s - 4: 
+ 2 
■I + .s! 
I O ’ f A s — 4 ! ^ i + s + 4 ! 
+ p'9s--i -1- P'A-+i--— 
^ + s ! 
i + si 
It will be observed that if .v is equal to i or i — I the terms in in ^i^cl qs + ^ 
disappear; and if .S' is equal to i — 2 or i — 3 that in (js+i disappears. This agrees, 
as it should do, with the vanishing of and ^ when the order is greater than 
the degree. 
