790 
MB,. J. W. L. GLAISHER ON RICCATI’S 
Transforming the integral in (12) by assuming x——, it is seen that (12) is true also 
when i — a negative integral; so that this formula is true when i— any integer. 
The same transformation shows that in general, if <£ is an even function, 
| x~ zi ~ 2 (f)(^x — C ^jdx=cr 2i ~ 1 ^ x 2i (f>(^x — ~^jdx. 
Putting 2 i=n — 1, the formula (12) assumes the form 
[ of 1 
Jo 
—I /y —x - 
r 'dx- 
v 7 ’ 
\16aJ 
which is true when n— an uneven inteo'er, the series being continued till it termi- 
nates of itself. 
23. The investigations of the formulae (9) and (10) given by Cauchy and Boole 
are only applicable in the case of i an integer, and do not indicate what the formulae 
become when i is unrestricted. A method, however, which I have employed in the 
‘Messenger of Mathematics’ (vol. ii., 1872, pp. 78, 79) to prove Boole’s formula, and 
which depends on direct transformations of the integrals, leads to the general theorem. 
We have 
| x n (fj(x — -jdx= j x’'(f)(x — a ^jdx 4- j x n <\)[x — ( -jdx ; 
and if we transform the second integral by assuming x= x -, , then, since </> is an even 
function, we find that the original integral 
= lA x ~i 
_i 
n + 1 
x“-\- 
yti + 1 
0 n + 2 
dx 
ci\ d 
X ~x)jx 
x 
,n +1 
7 «+l 
dx. 
Now transform this integral by assuming x — =v; we thus have x=^{v± \/( y2 + 4a)}, 
and, taking the upper sign, the integral becomes 
d 
1 
n +1 
r + v / C 2 + 4«)l " +1 [ — v + v/C 2 + 4a)|" +r 
2 
■ 
dv. 
If n is an even integer, the quantity in square brackets 
v + v 7 (r 2 + 4 a) 1 " 
+1 
1 
+ 
v— v / C 2 + 4«)l 11 
+i 
2 
n — 2 cr 
i 
= <’" +1 1 + (» + 1 ) i + (« + 1 ) V =-,+(«+ 1 ) 
. (n — 3)(?i“4) cl" 
■.in 
; 6 . . . -f (n+1) 
this expression containing \n -\-1 terms. 
