MR, W. M. HICKS ON TOROIDAL FUNCTIONS. 
G 1 7 
II. 
ZONAL TOROIDAL FUNCTIONS. 
5. In the case where the conditions are symmetrical about the axis, (p is independent 
of iv and is of the form 
cf>= aJ 
cosh, u —cos v 
sink u 
Zxfj n cos [nv-\- a) 
where \p u is the general integral involving two arbitrary constants of the equation 
From the potential of a ring, at the end of the last section, it is at once seen that 
a particular integral for space, not including the critical circle, when n= 0 is 
i// 0 =^/sink v\ 
j 0 
dd 
y/ cosk u —sink u cos 9 
In the same way, calling the potential of the ring <p, it may be shown by finding 
b(f> 
bz 
- that 
dd 
l/q "v/shlf cj ^^ cos ] 1 u _ u CQS 
From analogy with this we might assume 
_ _U dd 
v/smh U ) 0 (cosk^—sinh u cos 0)p 
2 n + 1 2n — 1 
and by substituting we should find it possible by putting p— —-— or ——-— to satisfy 
the equation. But the following, by making use of theorems already proved for zonal 
harmonics, seems to be more direct. Putting, in the differential equation. 
xp= \/sinh u. P 
— s -fcoth u— — {n — i)(«+i)P=0.(9) 
2n+ 1 
whence it is at once evident that P„ is a zonal spherical harmonic of degree —- , 
LJ 
with a pure imaginary for argument. Heine, in his ‘ Handbuch tier Kugelfunctionen,’ 
MDCCCLXXXI. 4 L 
