91 
1920-21.] Hypergeometric Functions of Two Variables. 
where W is of the hyperparabolo'id type, the connection between this 
function and the hyperspherical polynomials being exactly the same as 
that between the two one-variable functions considered. 
Chapter III. 
HYPERCYLINDRICAL FUNCTIONS. 
Another example of the use of one of the confluent hypergeometric 
functions for solving Laplace’s equation is given by the introduction of 
hypercylindrical co-ordinates. We shall first consider the problem in 
four-dimensional space, and afterwards generalise it. 
The change of variables 
x = p sin # sin </> 
y = p sin 0 cos </> 
z = p cos 0 
t = t 
defines what may be called hypercylindrical co-ordinates, the hypersurface 
p = const, being an hypercylinder with its generatrices parallel to the /-axis, 
and with the sphere 
X 2 + y 2 + Z 2 = p 2 
as a basis in the xyz space. It may therefore be termed a spherical 
hypercylinder. Laplace’s equation is then 
AU = ^ + 1S + 
1 
0 2 U + 0 2 U + 2 0U + cot(9 5U_ 0 
0/o 2 p 2 dO 2 p 2 sin 2 0 dcj> 2 dt 2 p dp 
dO 
A solution may be obtained by taking 
U = ef* cos 0), 
the two-variable function Up which we shall call a function of the 
spherical hypercylinder, or more briefly hypercylindrical function, being a 
solution of 
d 2 u 1 
dp 2 p 
2 86> 2 
2 dl\ cot# 01T 1 
p dp p z dO 
p 2 sin 2 0 
or, by putting cos 0 = go, 
(A) 
2 0 2 U Xj n 2 \0 2 U 1 j 0 0U x 0 0U X , 2 o TT 
P 2 — ^+(1 -o> 2 )^ + 2p— i- 2w—± + p. 2 p 2 XJ 1 - 
op L 0 (t) Z op 0(0 
Ui = 0; 
,U!=0. 
Let us denote by U 1 / (p, w) a solution of the same equation in which we 
gave to v the value zero. The product 
TJ' = e flt JJ 1 '(p i w) = e^U/(p, 0) 
