624 
MR. W. M. HICKS ON TOROIDAL FUNCTIONS. 
in the same way as P„ (for space outside a tore) was obtained from the potential of a 
ring, so also may Q /t be determined from the potential of a point at the origin, for 
space not containing it, i.e., for space within a tore. For the inverse distance of a 
point from the origin is 
1 
a 
0 —cos v 
C + cos v 
Hence 
— /„]■ — - -=S(AJ? M +B„Q,) cos (nv+a) 
v 7 C + cos v v v ' 
Now, firstly, since this is to be finite throughout all space not including the axis 
A„= 0 . Also it is clear since C> cos v that 1 V / C + cos v can be expanded in a series of 
powers of cos v, and therefore in a series of cosines of multiple angles only. If this be 
" done the coefficient of cos nv must be B„Q„. 
Hence by Fourier’s theorem 
7T 
9 
cos nOcW 
C - 4 - cos 0 
If we define Q /t so as to make 7 rB„/ 2 = (— l) H \/ 2 , then 
a/2 Q«= (-)“[" 
cos nddd 
J o -v/C + cos 0 
cos n6 
•' o 
v^C—cos 0 
dO. 
(23) 
We will now show that this expression for Q u agrees with the former one. Inte¬ 
grating by parts and dropping ^/2 as unnecessary in the sequence equation 
n r\ / U sin n0 sin 6 
2 «Q„(-f — - 7 -- - Q -.d0 
v ' J n (C+ cos 0> 
q (C + cos Q)' 1 
also 
! o (C + cos 0) 
=c( 
Jn C 
COS 710 f K COS 110 cos 0 
J o (C + COS 0)* J o (C + COS 0)5 
0«-H)Q»(-)"= C 
n COS 710 
o (C + c °s 0) 
H-f 
y cos (?i+1)0 
J o (C cos 0) • 
(2 n— 
i)Q«(—)"=—cf 
J o 
cos ii0 cos n —10 
(C + cos 0) ‘ J q (C + cos 0) J 
Hence 
