EQUATION AND ITS TRANSFORMATIONS. 
785 
It is however more convenient in the first place to consider the definite integral in 
the form 
y— I x l e * dx, 
which is obtained by transforming (5) by the substitution x’"=x' 2 , for the integral (5) 
thus becomes 
2 bz m 
— ~ & dx. 
J o m 
Comparing (6) with the standard form (4) of IliGCATi’s equation in art. 17, we have 
2 1 
m—2q, so that —=-=n, and ndb=a 2 , 
m q 
Let bz m =od ; then z = b~ hl a H , and we see that the definite integral 
satisfies the differential equation 
dhf 
do? 
n—ldy 
a dot. 
— 42/= 0. 
The value of this definite integral is therefore of the form AU 1 -1-BV 1 , where Uj, Y l 
are the same as in art. 16, a being substituted for x and a put =2 : viz., writing 
M=l — 
N=l-f 
l 
n — 2 
1 
n + 2 
( 2a ')"^(%-2)(a-4) 
^ 2a ^^"(% + 2)(a + 4) 
(2a 2 ) 2 _1_ 
2 ! (n—2)(n— 4 )(% — 6 ) 
(2« 2 ) 2 1 
2! (% + 2)(?i + 4)(7i + 6) 
(2a 2 ) 3 
3! 
(2a 2 ) 3 
3! 
&c., 
See., 
then 
x 1 e~ 1 ~ x “ dx— AM + Ba ,! N. 
0 
Suppose n positive, and put a=0 ; we thus find 
and therefore 
MDCCCLXXXI 
A = [ x 11 l e x \lx=\Y(\n), 
j 0 
f x~ l e 1 ~ x ' 1 dx=^Y{bn)M.-\- ot n <f)(n)'N. 
j 0 
5 i 
