636 
MR. W. M. HICKS OK TOROIDAL FUNCTIONS. 
Hence 
A=1 and P-„= 
cos 
= 'o(°- S 
cos mddd 
2/1 + 1 
— S cos 6) 2 
Returning to the general integral, since u, u enter symmetrically, and since if 
u^u, u>u, it follows that 
p cos mw cos nvdwdv 
0 •’0 
y/CCf — cos v — SB"" cos w 
■ LP/b.ji' Q m.n 01 LP mji- Qa».« 
according as u^u, where L is independent of u or v!. Hence L may be determined 
by giving particular values to u or u. Suppose u r at the origin then 
RQ;h.//— hm„/_Q 
cos mvj cos nvdwdv 
o j Q y/CC / — cos v— SS' cos w 
C 77 cos mOdd 
2 / 1+1 
(O'-S' cos 0)~2 
To find the value of this expand the expressions under the integrals in ascending 
powers of S', which is ultimately to vanish. 
Then if 
p < m cos mw cos^ ivdw =0 
*'o 
p~vi j cos mw cos"' ivdw= j- 
p>m the integral is finite =1 
Hence 
wher 
re 
[" , c* cos mw 
cos nvdv 
— Jo_ Jo ~ cos v 
PQ/«.«= lim 
SS' cos w \ m 
m ' CC'- cos v 
+ 
-} 
Q' 2 n+i cos mO (/3„ 
2 Jo \ 
S'™ cos™ 6 
C' m 
f • • • 
w 1.3.5 . . . (2m—1) 
,= CO. of x m in (1— x)~ s — -—-p- 
P _ M (1 x f¥__ <?* + l)(2n + S) • • • (2n + 2w—1) 
2 pm 
LQ„,„=“-s4 
Pm 
J c 
cos nvdv 
2 ../ + 1 
(C — cos r) 2 
Hence 
