92 
Proceedings of the Royal Society of Edinburgh. [Sess. 
is a solution, independent of 0, of Laplace’s equation, and therefore a 
zonal harmonic function. We shall say therefore that the function U/, 
solution of 
i&U’ n fell' 0U Y 9 SUf 2 2TT ' O 
( B ) p ap 1 +1 1 )-^ +2p ^y “ 2o>-^J- + /xV u i 
is an hypercylindrical zonal function. 
Such a function is readily found by considering our confluent function 
Z = E 2 (a, /?, y, X , y), 
which, as we said, satisfies the system 
jx(l “ x)r + ys + [y - (a + ft 4- 1 )x\p - afiz = 0 
+ xs + yq — z = 0, 
and therefore the single equation, obtained through elimination of 6', 
x\x - l)r + y 2 t + yqy - [y - (a + /3 + VfX^px + a/3zx - zy = 0. 
Let us now, in this last equation, make the change of variable 
x = 
y = \yf 
and substitute for 0 a new function f defined by 
Let us then give to the parameters the special values 
We find 
a=l, £ 
= 1 
r=|. 
(1 -«l + ^ +2 ^- 2 f- 4 ^= 0 ’ 
M 2 
an equation which becomes identical with (B) if we take A= — ~ ; so that 
the function 
or 
1 „ 
, 2 n 2 
W<* e)= «te u A 1 ’ i; §; 1+tan2e ’ _/ v 
is a zonal hypercylindrical function. 
The research of a complete hypercylindrical function is now exceed- 
ingly simple, if we make the following remark : if U/ is a solution of (B), 
a solution of (A) will be 
i 
U 2 (p, ti,) = (l-C0 2 ) 0-^U/ 
m d m 
(just as P n (a?) = (l — x 2 )- — m P n (cc) is a solution of the associated Legendre 
dx 
equation, and a complete spherical function, while P n is a zonal one). 
