EQUATION AND ITS TRANSFORMATIONS. 
797 
whence, differentiating with regard to /3, 
f” a r lXI rl 
j ar* + i e- px *’dx= 
Now — n+l lies between 0 and 1, and putting a = 0, we have 
giving 
x -* +1 e-v'=%P n - 1 r(—%n+l)=—%nl3F t - 1 B, 
B= -; r UU+i)=lr(-U)- 
Thus, if n lies between 0 and 1, it has been proved that 
[ x u ~h~ ac '-^dx=^r(^n)a.- in ll ll -\-±T(—^n)l3 h 'K n , 
J o 
and this equation can be readily shown to be true for all values of n by differentiating 
both members of it any number of times with regard to a or /3. 
For, differentiating with regard to a, 
00 £ 
x >l+l e~^~x*dx=\T(\n). \n a^' i_ 1 H a+ 3 ”|r( 
o 
=ir(I»+i)«" 4 “- 1 H, + 2 +ir( 
and, differentiating with regard to /3, 
fV s e-“•-^=jr(|n) 
Jq ^ 
=ir(i«-i)a-^iH H . s +ir(-in+l^K^. 
If, therefore, the formula (7) is true when n—r, it is true when n—r± \ ; and it 
has been proved to be true for all values of n between 0 and 1 : it is therefore true 
for all real values of n. 
It may be remarked that whatever value n may have, the integral is never infinite : 
so that the differentiations with regard to a or /3 are always permissible. 
_i 
h n ) 
2/3 
ln+1 
n-\- 2 
K 
—\n— l)/3 u+1 K n+il , 
