MR. W. M. HICKS ON TOROIDAL FUNCTIONS. 
631 
which since 
becomes 
Hence 
If we choose 
then 
P" I ^p' _ ^^ p ft p _ A 
x n\ g- 1 - n ^ x n g2 J ' n —^ 
P„={4m--(2, i -l)-|A,._ 1 B„P„ 
1 
^-n —— : f.-y 
m + n — l)(2m—n+ 1) 
— 2 (mfi-w) — 1 
tU =2(m— n) — 1 
Hi/. 
these conditions are satisfied, and. the formulae agree with those found for the zonal 
function when m = 0. Hence 
2SP m .n — (2mH-2w.fi- l)P m/A+1 — (2wfi- l)CP ilu 
2SPfi,,= (2m-2n + l)P„ llB _ 1 fi-(2n — 1)CP„,„ 
From this there follows at once the sequence equation 
(26) 
(2mfi- 2n fi- 1 )P m ., (l+1 — 4wCP m , t +(2w— 1 —2m)P, il . H _ 1 =0 . . . (27) 
In this write 
Then 
P = 
-*• on.'ii. 
whence, if 
(2m+2w-l)(2m+2w-3) . . . (2m+ 1) WM 
ri (2 n —l) 2 —4m 2 
1+1 2w(2« —° 
(2w-l) 2 -4m 2 
Qon'n. 
and a l)i n , are the same functions of c mM , &c., as a d , a / „_ 1 are of c n 
p _ (2w-2) (2n — 4) ... 2 _ 
(2m + 2n —I)(2 to + 2;. 5 . —31 . . (2m f I) 
{ ^»i.s.Pm.l 2®^ m.n— lPm.Qj 
(28) 
These formulae hold for the two particular integrals P„ ltW and Q m-W , and they 
express the tesseral function of any order and rank in terms of sectorial functions 
