1920-21.] Hypergeometric Functions of Two Variables. 93 
Therefore 
Uj = sin v 6 
d v 
0 (cos 0) V - 
is a complete hypercylindrical function. But the differential which occurs 
in this expression is easily reduced. Let us, for brevity, put 
/*V_ 
We have, considering the expansion of 3? 2 , 
(j, m) 1 u n 
and 
'V ZZ (! 
0'TTf / "V (£> m )(2w + 1, v) 1 u 
(-§, m + n) ^m+i ^ j 
Soj" ~ ^ ^ 22/ + to 2m +*'+i w.!’ 
and a rather simple transformation shows that 
(J, m)(2w + l, v) = 
+ 1, m )( n +b m 
m ! 
so that we have 
0 W =( _ ir Avy 
2 + b + i» m ) i 
do) v 
(f-, m + ra)m! o) 2m n ! 
= < - 1 )"d+ H2 (i + lj i + 1 5 b “} 
The function 
Ui( P) 6) = 
tan,, 0 ^ fv 
cos 0 
5 + 1, £+1; I; l+tan*«,-££ 
is therefore an hypercylindrical function, Laplace’s product for hyper- 
cylindrical co-ordinates being 
u = cos v<y^H 2 (| + i,|+i; I; i + e , - 
The same problem in {n-\- 2)-dimensional space leads us again to the H 2 
function. If the Cartesian co-ordinates are denoted by x v x 2 , . . . , x n+2 , 
hypercylindrical co-ordinates will be defined by 
= p sin 0 sin (f) 1 sin <jf> 2 . . . sin <£ n _ 2 sin ^ n-1 
x 2 = p sin 0 sin ^ sin <j> 2 . . . sin <f> n _ 2 cos ^ n-1 
x s = p sin 6 sin cf) 1 sin cf> 2 . . . cos cf> n _ 2 
x n = p sin 0 cos 
x n+ i = p cos 0 
