1920-21.] Hypergeometric Functions of Two Variables. 95 
with respect to C^the same part as P™ to P n ; and we have in particular 
C* = P*. 
fJL A 
It is easily seen that the function CT (cos <p) satisfies the differential 
equation 
/72/w rim u( n _L_ O.ir 1 \ 
= 0, 
d 2 u , 0 , , du 
w v 
A(X+2,)ij^ + ^- 1) 
sim (f> 
so that there appears a way of solving our above equations by taking, 
in general, 
i - 1 
fii&n-i) = C (COS <fr n -i). 
v i — 1 
The last equation, which contains only 0 and p, is found to be 
A 2 0 2 U, , 0IL , 0 2 lL , , 1N , /j 0U-. TT 
, 2 n 2 , %-tK-! + n - 2) 
P sin 2 0 
We can then easily verify, as we did in the case of four dimensions, 
that a solution of it is 
TT tan m n-i ( m n _ x + » + ra - l . w + 1 . , 2 A k 2 p 2 \ 
— 2 — * 1+tan ~~r)’ 
so that Laplace’s product in hy percy lindrical co-ordinates and (n-\- 2)- 
dimensional space is 
~ ^ tan mn_1 0 
U = COS ^(oos <t>n-2)Gli3, ^(cos <£„_ 3 ) . . . C^_ tVi (™ 0i) cogn _ 1 
x S 2 ( m -‘ + ” , ro "- 1 + m ~ A ; 1+1 • 1 + tan 2 . 
If we seek a solution independent of the 0’s, which will therefore be a 
zonal function, we equate all the m’s to zero, and obtain 
IV(P, 0) = 
1 ^ fn n— 1 7i + l 
cos 
Z-2 n 2 
; 1 + tan 2 0, — — £ 
2 \2’ 2 ' 2 
Let us take in particular n= 1 : we are in three-dimensional space, and we 
obtain a cylindrical zonal function which is here 
U^p, 6) = S 2 (i, 0; 1 ; l+tan 2 0,-^!); 
but this is equal, through the expansion of % 2 , to 
/ &2 p 2\n 
zv V. 
w (1, n)n ! 
which is precisely Bessel function J 0 (&p) ; an obvious result which, however, 
affords a good confirmation of the preceding theories. 
