MR. W. M. HICKS ON TOROIDAL FUNCTIONS. 
637 
This at once gives us an expression for Q />u „ viz.: 
Q m.i/ 
cos nvdv 
2 ill +1 
(C— cos v) 2 
where M is some constant depending on m.n. This has now to be found. If we 
write 
S cos nvdv 
2iit+T 
_ 0 (C - cos v) 2 
it is easily shown that 
(2 n -\-1 —2;n)U m .u +\— 4/iCU w- „-f-(2 n — 1 -(-2m)U OTiW _ 1 =0 
This will agree with (27) if 
N tt (2w + 2»,-l) . . . (2m + 1) n 
m ' n (2 n -1 - 2m) ... (1 - 2m) Hm ' n 
Hence 
and 
(2m-l + 2?t) . . . (2m+ 1) „ 
(2?i —1—2m) . . . (1—2m) 
C dv 
Q W . 0 =NS" 
2m +1 
(C— cos v) 2 
where N is a function of m only. 
Here again this is found to satisfy 
(2 m-\- 1)SQ W+1 . 0 —4mCQ w . 0 +(2»i — 1)SQ,„_ 1<0 —0 
which agrees with (32) if 
N= (— )'"N 
and 
n ft f’ r ^ v 
^°- 0_JN Jo7^hsT 
But from the known value of Q 0 0 we see that N=l/\/2. Hence 
Qw///— 
(-)» (2?t-l—2m) . .. (l-2m) c< 
2 (2n — 1 + 2m) . . . (l + 2m) 
cos n6dd 
2 ni +1 
. (C — cos 6) 2 
(34) 
