632 
MR. W. M. HICKS OH TOROIDAL FUNCTIONS. 
and tesseral functions of the first order and same rank. In the same way as was 
proved in the case of zonal functions, it may he shown that 
P O — P O 
m.n+\ y Xim.n x + l 
(2ft—1 —2m)(2ft—3—2m) . . . (l-2m) 
(Pm.lQm.O Pm.oQm.i) • • • (29) 
also that 
(2ft +1 + 2m) (2ft — 1 + 2m) . . . (3 +2m) 
(2ft —1 —2m) . . . (1 —2m) P, u Q»».n- p ^.oQ«.i 
P' o —P O' ■ 
1 m.n^xim.n *- m.n~ 
(2 ft+ 1 + 2m) . . . (3 +2m) 
2S 
• (30) 
13. In the same way as relations have been found between successive orders of 
toroidal functions, relations may be found between successive ranks. 
r Not putting the order n in evidence, write 
\fj m+l =fxfj m -\-(j) \fj' m 
Proceeding as before it will be found that f <j> must satisfy the equations 
/ 
2m +1 „ 4m 2 — 1C 
2 S 3 
S 
—J 
Ui</> + 2w 
4m 2 -1 
4S 3 
<f> = 0 
1 
which are satisfied by 
leading to the relations 
</j=A, /= 
2vi +1 . C 
P m+\ —A> fl ( P m WlgPrn 
In precisely the same manner it may be shown that 
P„_ 1 =B,(P'„+m|p„ 
and that 
If we put 
then 
A P, — 
1 — 
A„,= 
4 
4ft 2 — (2m +1) 2 
o 
B„,= 
2ft + 2m+ 1 
9 
2ft— 2m 1 
