G12 
MR, W. M. HICKS ON TOROIDAL FUNCTIONS. 
Here since u, v are conjugate functions of p, z 
so that 
***** 
0 
(6p\*(6 P \*_ 1 1 
\ bit) \bv) V 
(say) 
b-yjr b^yfr 1 f 
bid bv 2 Vf 1 
( 2 ) 
By putting -~=0 we get the equation for functions satisfying conditions sym- 
metrical about an axis which by an obvious analogy may be called zonal functions. 
In general, put cos (mw-\-a), then \Jj' must satisfy 
b 2 \Jr' b-yjr' 
4m 2 — 1 
~W~ 
f=o 
When u, v are functions such that 1 /(p|) 2 is of the form 4(/(zt) + F(r)) J it is possible 
to obtain solutions of the form x/j=XX„ ui Y„ cos (mw-\-a) where 
^"=(4m*-l)/(«)X+»»X 
'W= (4m ! -l)F( B )Y-» s Y 
which are such that X,„. M are constant when u is constant and Y»,.« constant when v is 
constant. 
As an instance of functions satisfying these conditions we may take the elliptic 
co-ordinates 
p=ci cosh u cos v z—a sinh u sin v 
Here 
1_I__1_ 
/r£ 2 cos 3 v cosh 3 u 
And the equations for the functions X, Y are 
cF~X 
du 3 
+ 
/ 4m 3 -l 
\4 cosh 3 u 
~\~n~ ] X—0 
d~ Y 
ifo 3 " 
/ 4m 2 — I 
\4 cos 3 v 
± rA Y = 0 
