77 
1920-21.] Hypergeometric Functions of Two Variables, 
take 
(1 - tx - vy + vtx)~ a *= ( - 1)~“[1 - (1 - tx) - (1 - vy) - vtx\~ a 
whence 
( «, m + n+p ) t 
m n p 171 \ 71 \ p \ 
r, = (-i)-222* fl 
(a, m + n + p)(p, p)((3', p)_ 
m\n\p\(y, p)(y - /?', p) 
and 
3> 
F( - m, /3 +p ; y - p' +p ; x) 
F (~n, /¥+p, y+p, y) 
y-fi + p, y). 
All these expansions show the intimate connection between these functions 
and the similar one-variable functions. 
It is easy to show also that an important relation exists between <i > 1 
and and that, in fact, they always reduce to one another. Let us start 
from the expansion which we gave for <L 1; in ascending powers of x ; then, 
using the relation 
<I>(a + m, y + m, y) = e y ®(y - a, y + m, - y) f 
we can write 
Z (a,m)(i3,m) x x m 
^ -y $(y-a ; y + m, - y) 
and, comparing with the expansion for S 1? 
$ i( a ; P ; y ; y) = e y %i(<h y - a ; P ; y ; x % y)> 
which is the relation in question. 
m ! 
Chapter III. 
DIFFERENTIAL EQUATIONS SATISFIED BY THE FUNCTIONS. 
The seven confluent functions satisfy partial differential equations of 
rather simple forms, which it is easy to obtain, by confluence, from the 
four systems of equations found by Appell for the F functions. 
W riting 
dz dz d 2 z 
ay -a. S?-r, etc., 
we find that the system for the function flq is 
rx( 1 - x)r + 2/(1 - x)s + [y - (a + P+ 1 )x~\p - pyq - apz = 0 
l yt + xs + (y - y)q - px - az = 0, 
