622 
MR. W. M. HICKS OX TOROIDAL FUXCTIOXS. 
We first show that this with a modification satisfies the equation (9). For putting 
cW 
Q ,1 1 2n + l 
J n (C + S cosh 6) 
(19) 
dQ 
du 
<C‘l 
2a+l l _! 
2 J.(c- 
S + C cosh 6 7/1 
+ S cosli 6) 2 
d~Q„_ /2n + 1\2^ 2%+l 
did' 
! 2n+ir d 2,1+3 
Qu-— 1 ] sinlr *|(C+S cosh 0)~ ^cW 
2n+ l\ s n 2?i + ll cosh 6dd 
2 ) 2S 
2/1 + 3 
(C + S cosh 6) 2 
d~Q n , dQ f2n + l\~ r . 2?i +11 CS+ (O— 1) cosh 8 in 
— + COth Q, jiT \ - 
•JO 
(C + Scosh#) 2 
4n 3 —1 
Q, t 
Here also as in the case of the P functions it can be easily shown that 
2S 
2n + l 
dQ„ 
du 
—Q«+i 
-CQ„ 
2S 
2»-l 
dQn 
du 
=CQ„-Q„_ 1 
and 
(2 n + 1)Q„ +1 - 4?iCQ* + (2 n - 1 )Q,_ 1 = 0 
Hence as before 
_ (2n — 2) .. . 2 / 
Q/i (2il— 1' o( a /di o a «—l^o) 
where 
H 
dd 
Vi = 
o \/C + S cosh 6 
“ dd 
1 J o (0 + S cosh 6)* 
