494 
PROFESSOR O. H. DARAVIX OX ELLIPSOIDAL HARMOXIC AXALYSIS. 
I now introduce a further abridgement and T^'^ite 
(., _ 1) i (.^ + 1) 0 4- 2) 
s2 - 4 
( 32 ), 
or shortly T. 
Then, after reduction, I find 
^ + 4S + :3 + T] P + . .. 
Accordingly tlie coefficient of P'^ in P' is 
1 + P(X + I) + ^/3Ht + 1) - -,W[- + 22 + 41 + 3 + T 
- i/3-^ i 
\Js — l)(s - 2) 
- I ' 
: 3T + ^ 
{e, g + 1} {t, -s + 2} 
(s + l)(s + 2) 
^ ' (-5-4) {%, s] {t,s - 1} , s + 2 1 XI 1 ^ X • xi 
But q = ----, q , + o = TTZ -77; f^^^d the fast term in the 
^ - 8s(s — 1) ’ + - g_ 5(5 2)’ 
above expression will be found to be ecjual to + — 1). Thus the coefficient 
of in the development of is 
I + 4/3(X + 1) + + 22 + 42 + 3 - T); 
but the same coefficient in is unity. 
Therefore 
d. = 1 + i/S (S + 1) + -,7/3-(2V + 2= + 4S + 3 - T) 
^ i 
cv = 1 - i/S(2 + 1) + 2V + 32^ + 42 + 1 + T) 
= I + ^ (2 + 1) + il3’- (2V + 32^ + 82 + 5 - T) 
= I - /3(2 A I) + Ih- + 522 _p 82 + 3 + T) 
• (33). 
'I'he squares of this constant and of its reciprocal are given because they will be needed 
at a later stage. 
We next consider the cosine and sine functions. 
As far as fi- 
= (1 — /3cos2(f)y = I — ^/3co8 2<f) — -^q/3'{I A cos 4^). 
