EQUATION AND ITS TRANSFORMATIONS. 
817 
then, denoting for the sake of brevity a 2 ar + , by w, 
dv 
da 
whence 
and therefore 
f* ou 
= — 2 ax 2 e~ w dx, 
Jo 
— ^ x~e~ w dx 
la da ,/ J ^ 
Again, 
and 
therefore 
whence 
1 A 
2 a da 
[ x %i e~ w dx 
J n 
•■) r co 
dh; 
da, 3 
- = (— 2x 2 -f ia 2 x 4 ) e ,v dx, 
J o 
[ 4crx% w dx— 
J n 
■ e a °' x '‘2x B e x' 
00 *00 
+ Jn 
0 •'o 
e - aV —(2a? 3 c z j )<Ac 
Ctdi/ 
* CO 
= | (6x 3 4- , l6 2 )c _ '''c?a’ 
•'o 
d 3 
da : 
1=1 (4x 2 +46 2 )e~ H ’<iay 
Jo 
da 3 
(51). 
46 2 W = 4 
are !C cLc.(52). 
If instead of the integral v we start with the integral 
Vi— I x zi e~ w dx, 
J o 
we have 
~= f°(- 2x 2i+3 + 4 cM i+i )e-”dx, 
clcc* J q 
and, integrating the second term by parts as before, we find 
C jk= ( {{ii J ri)x 2 i+ 2 -\-ib 2 x 2 i }e~ w dx, 
so that 
(~— 4D, = 4(fH-1) f x 9J+ ' 2 e~ v: dx=i(i+ l)u +l .(53). 
MDCCCLXXXI. 5 X 
