EQUATION AND ITS TRANSFORMATIONS. 
789 
To deduce the value of the integral 
< p(x) = e~ x \ then 
•Q° a 2 
x l e~ xl ~ x "dx from Boole’s formula, let 
o 
x 2i e~ x 
'dx=\Y{i+\) = l f 
27 — 1 _ 
2 ’ 
and therefore the coefficient of a 1 m 
y /r rr (2m+l)(2m + 2) . . , (7 + m) , 3 
2 (7 + m)! 2 ‘ 2 
% 
m— 1 \/7r (7 + to) ! 1 
Thus 
*A-ir dx= ^L 
whence 
x e v 
■ . (7 + 1)! ai- 1 (7 + 2)! a 1 3 
9 i^ + 77 + A 
(7—1)! 1! 2 3 ' (7—2)! 2! 2* * ' ' 
\A r . ( 7(7 + 1)/ 1 \ (7 —1)7(7+l)(7 + 2)/1 
\/<r-A dx= Y^ a i\ i+=^- 
2.4 
m)!m! 2 2m ' 
, (27)! 11 
+ 7! 2 2i J ' 
(in. 
\ 3 1 
j +&c. >e 2a . 
(12). 
the series being continued till it terminates of itself. 
This formula is in effect that given by Cauchy (‘ Exercices,’ loc. cit., p. 55) for 
the evaluation of the integral. It had however, as Cauchy himself remarks, been 
previously published by Legendre in vol. i., p, 366, of his ‘Exercices de Calcul 
Integral’ (1811). Legendre, whose method is quite different to Cauchy’s, adds 
that Euler, in vol. iv., p. 415,* of his ‘ Institutiones Calculi Integralis ’ (1794) 
mentions the integrals 
l+a* y 0 _ s _ i+z s 
x ¥ e 2nx dx, x % e 2nx dx, 
jo Jo 
which correspond to the particular cases i= — 1 and i =—2 of the integral in (12), 
as apparently not admitting of evaluation by known methods ; and he gives their 
values. 
If in the series in (11) the terms be written in the reverse order, we have 
2 i —z 
x e 
'A-dx = 
\Ar (27)! 
9+VI 
1 +2l 2S “ 
7(7—1) 
27(27-1) 
2%- 
2 ! 
e 2a 
which agrees with (8) when since 
ir(i+i)=^ — 
. (27 — 1) yir (27)! 
2 i 2 2 3i 7! * 
* ‘ Supplementnm V.’ ad tom. 1, cap. viii. 
