PROFESSOR G. H. DARWIX OX ELLIPSOIDAL HARMONIC ANALALSIS. 
4H‘2 
Let us write q\ ± 2n = (•^' db q, ± 2n- The qs are not now the actual coelEcients 
of any P-function, hut we shall see that they are determinable by almost the same 
relationships as those already found, and therefore the notation is convenient. 
A¥e now ha's'e 
= n[p,.s'P^ + tlB'‘q,_2n{s — 2?i)P’ t^’'q, + 2n{s + 2??)P + -"]. 
Applying the operation i//, to P'' and equating — ^ (P') to zero, we have 
S8n (s — n){s — 2?^) p, _ ,, 1 ?" “ (.5 + n) [s + "2n) q^ + 2 ;J^' * 
+ ?.[(« + + - 20 - 5 P’ + \i, s} {t, .s‘ - l}{s - 2)P-^^] 
+ ^/3'’q,_2n[{s ~ 2u + 2)p-2" + ^ - 2cr(s - 2n)P^-'" • 
+ {i, s — '27 i\ {t, s — 2n — 1}(^< — 2n — 2 ) P"'“ 
+ + 2 „ [{s + 2u + 2) P'- + 2o- (5 + 2n) P’ + 
+ {i, s + 2n] {?:, .9 + 2n — 1} (s + 2n — 2) p + ^n-sj 
= 0 . 
This is the same equation as before, if we replace ^P^ by Pd As we may equate 
coefficients of tP^ to zero (instead of coefficients of P''), we obtain the same equations 
for the g’s as before. 
A certain change must, however, l)e noted with respect to the beginning of the 
first series, which determines the end of the first continued fraction. 
We previously wrote P^ for [f, 0] {f, — 1] P“^ and i {i + 1) P^ for {i, 1} [i, 0}P~h 
But the corresponding terms will now he {f, 0} [i, — 1] (— 2)P“~ and \i, 1} [i, 0] 
(— I) P~fi and these are equal to ~ (2P~) and — (1 . P^). 
Hence it follows tliat when s is even (EES, OES) 
-'h - {ip} {i 1 2q, _ - {/, 4} {i, 3} 
Yo — /3a- ’ s- — 4 — ■ 
Tile po fias disappeared from tlie latter of these, and thus the continued 
raction is independent of Pq. This is correct, since whatever value (short of infinity) 
p,-) may liave p'q, being equal to OpQ, vanishes. Hence the continued fraction is 
docked of one term and ends with 
- lb 3} 
_ 4 - /i^O- ■ 
It is important to note the deficiency of one term in the fraction, since it indicates 
that when s ~ 2 the first continued fraction entirely disappears. 
When .9 = 0 there is no function of the P form, so the question of interpretation 
does not arise. 
