MR. W. M. HICKS ON TOROIDAL FUNCTIONS. 
633 
and when m —0 the P 0 .„ have the same values as for the toroidal functions already 
discussed. 
Finally then, 
2SPV«— 2mCP» W2 + (2n-f- 1 d - 2m)SP w+1 .» 1 
2SPV„= — 2mCP».„+ (2n+1 — 2m)SP w _ 1 .„ J 
from which the sequence equation follows at once 
(2m+2 n -f-1 )SP W+1 +4mCP W2 + (2m—2a — 1) SP M _ X =0 
If we write in this 
P — 
2m(2m —2) ... 4.2 
C\* 
then 
(2m + 1 + 2n)(2m — 1 + 2n) . . . (1 + 2»)\S 
, (2m-l) 2 -4#/S\ 2 
fn -I\2 -| ( r\ ) 'U'M—l — 6 
(2m —l) 2 —1 \C 
(32) 
By combining the formulae (26) and (31) it is also possible to obtain relations between 
order and rank together. For instance, from the first of equation (26) and the second 
of equation (31), we get 
(2m + 2» +1 )P«.„+! == (+1) CP m .n +2 SP' OT . M 
= (2 n-\- l)CP M —2mCP OT . ;2 +(2/id-l —2m)SP,„_ 1 ., i 
— (2 ii -\- 1 —2m)(CP, B . w +SP„ i _ 1 . ;2 ) 
with three other relations. 
The four formulae are 
(26a, 31a). P«.»+i—CP W .„—SP„ 2+1 ., t =0 
(26a. 31/3). (2n -\-1 -\-2m) P„ W2+1 —- (2n-{-l — 2m)(CP M .„d-SP w _ 1 .«) = 0 
(26/3. 31a). (2m-f-1 — 2w)(P^_ 1 — CP» 2 .„) — (2m+1 + 2w)SP OT+1 .„=0 
{26/3. 31/3). (2m+l-2n) P^_ l +(2m-l-h2n)CP w .„ + (2m-l-2»)SP w _ 1 .„ 
We are now in a position to reduce still further the relations (29), (30). 
For putting n = 0 in the second of (32a) 
(2m+ 1 )P?k.i— — (2m—- l)(CP OT# Q-f- feP» 2 _i.o) 
whence 
(2m +1) {P».iQ*. 0 “ P*oQ*.i} — (2m l)S(P m . f| Q w _ 10 P/«-i.oQ«.o) 
MDCCCLXXXI. 4 x 
I 
) ( 32a ) 
=oj 
