239 
As a simple case, if the variables x and y be connected by the condition 
x+yZl 
then will 
(7) .T(m) 
I-1,,m-1 af = D Dn), aaa a see 1) 
[Jeerne y”™ da dy = ® (e+ em) T(14l+m) 
It is obvious that, if the variables be connected by the same condition 
as before, and their number be p, we shall have for the result of the eva- 
luation of the multiple definite integral 
|]. . D(etytst...-p)arlymenr!... dedyds..., 
in its symbolic reduct form, simply 
T(J) 0 (m)0 (n)... 
A Ne r 
® (A, +4, + nt &e.) T(l+l+m+n+...) 
As a case of this last theorem, it appears that if the variables x and y be 
connected by the condition 
z+yZi, 
then will 
I'(Z) . 1 (m) 
Ci) Val -1 y,m-1 J, = Co) . ey 
[J Pty Baty dedy=O (Art An) BOT my 
There is no difficulty in extending these results to the more complicated 
case in which the variables are connected by the condition 
Seg Gfema 
(7.) It may readily be proved by the assumption x? =z, that 
WAS ea aoa 
“a(a+1)(a+2)...(a+9) 
1 
|, d= 2 ee de=3 
or, in the notation of Kramp, 
1/1 jv 
eee 
In the same manner it may be proved that 
2.4.6.... (29) 
(24+ 1) (2448)... (2441429) 
or, in the notation of Kramp, 
1 
ii (1 -— 2")? a" dx = 
Qa/2 
(2a + 1)e2" 
