MR. W. M. HICKS OH TOROIDAL FUNCTIONS. 
GIG 
r 
and 
(u-\-vi)(p J r zi)=a 
“P 
U= 
P" + r 
v— ■ 
az 
o . o 
I 
p + 
cou 
P o i ■> 
' u~ + v~ 
av 
die _ y _ _, 
chi =Z ^ = ^+? = ~a 
d : /r 4m~— 1 
Jui - ni , - ^*=0 
(8) 
It will be shown that between the latter surfaces and tores there is a similar rela¬ 
tion to that between cylinders and spheres, and between the functions to that between 
Spherical Harmonics and Bessel’s functions. 
4. The potential due to a ring of radius b, centre at (o.z) and plane perpendicular to 
axis of z, is 
bclO _ 
z') i + p 3 + 6 3 — '2bp cos 0 
n dd 
0 \/« — cos 6 
In the case where it is the critical circle 
O , 0 , o 
«=-t£±^=coth 
2ci p 
and here 
dd 
* o \ 
/coth u — cos 6 
In general the distance between two points is (z—z) : ~\- p : -\- p' z —2pp' cos (w — id), 
which expressed in bipolar co-ordinates becomes 
2 a* 
(cosh u —cos A) (cosh v !—cos d) 
, , 7 {cosh u cosh u —cos (v — v) — sinh u sinlr u cos (w — it/)} 
