788 
MR. J. W. L. GLAISHER ON RICCATI’S 
Sections, pp. 15-17) I gave the formula (8) with a brief indication of the method by 
which it had been obtained; this method is substantially the same as that just 
explained. As far as I know the general value of the integral had not been given 
before; although the value in the case of n = an uneven integer has been long 
known. It is scarcely necessary to remark that in (8) n must not = an even 
integer : this case is specially excepted throughout (see end of art. 1). 
22 . The case when n — an uneven integer is included in a general formula given by 
Cauchy in vol. i. of his ‘Exercices cles Math ematiq ues ’ (1826), pp. 54-56. He has 
there shown that if 
i being a positive integer, and <jj an even function, then 
n p | *'(®+l)p , (t—l)i(i+l)(i + 2 )p 
kbi— r Q\~ O! *21 4 , • 
■ • (9>- 
This is the case corresponding to a— 1 of a formula proved by Boole (Philosophical 
Transactions, vol. 147, 1857, p. 783), viz. 
f 
J ( 
i d\ , .m=i(2?n -(- l)(2??i -f- 2) . . . ._f „ ., , 7 , . 
x^[x--) dx =t m=o -^ X^<f>(x)clx . . (10). 
Boole’s formula may however be deduced from Cauchy’s ; for, replacing ^(x) by 
(p(ax), we have 
Q 3 j=j x zi <f)(ax— -jdx, 
and this integral, transformed by assuming x =y, becomes 
dx, 
in which, if w 7 e put a — b and replace a? by a, the expression subject to the functional 
sign becomes x —* 
& x 
* In the ‘ Messenger of Mathematics,’ vol. it, p. 79, I stated that Cauchy’s proof was not applicable to 
1 ct 
the more general theorem in which *-was replaced by x -. The error was corrected in a paper 
“On a Formula of Cauchy’s for the Evaluation of a Class of Definite Integrals” (‘Proceedings of the 
Cambridge Philosophical Society,’ vol. iii., pp. 5-12, 1876); this paper contains also the theorem, 
corresponding to Cauchy’s, in which 0 is am uneven function. 
