792 
MR. J. W. L. GLAISHER ON RICCATI’S 
and could not be rendered satisfactory without careful discussion and development. 
Such substitutions, however, very frequently give correct results, and it is generally 
interesting to examine whether, in any case that arises, the result so derived is true or 
not. In this instance also we thus obtain two new forms of the expression forming 
the right hand member of (8). 
i 2 
25. Transforming (8) by the assumption x—v n , putting m— -, and writing a for a, it 
will be found that 
C oo w / 
f e~ l ' m ~^dv= r(l 
o \ 
o( 2 «M 
(vi — 2)(Zm—2) 
(ra — 2)(2m— 2) 
b&c. 
e -a * 
(m + 2)(Zm + 2) (2«) 3 0 
(m-f2)(2m+2) 2! + < * C ' 
( 13 ); 
where, as before, if either series terminates through the presence of a zero factor in a 
numerator, the terminating series represents the value of the integral, and if neither 
series terminates, both are to be included. When one of the series terminates, that is, 
when m — twice the reciprocal of an uneven integer, the formula may, by taking the 
terms of the series in the reverse order, be written 
3 CL / 1 
-v m —- 7 \/7r 
e v m dv= x — a m 
m 
m‘ — 2 3 / 1 \ (m 2 —2 2 )(3 2 m 2 —2 2 ) / 1 \ 2 
1— "(-'*)+ "" " \l6« 
•mf.wn* 
_ _ 
J 0 m [ to 2 \16fl 
2 
Putting v=a.™x and a~ = orfi' } (13) becomes 
r 00 o i 
f - a?x m - 
e 
Jo 
— t(i — -']/3 71 {1 -4- — + 2 (2a/3) 
\ r^m + 2^' 1 (m+2)(2m + 2) 2! 
—&c. \e . 
"efcc=r( 1 + —W TO {l + // —”( 2 ay 8 )-f 
v m) I vi —2,^ ' 
(m-2)(3m-2) (2«/3) 3 , 
(m—2)(2m—2) 2! ^ '' 
(m + 2)(3m + 2) (2 a/3) 3 
-&c. je 3a 4 
If for example m=2, we have the well-known result 
26. The definite integral 
f 
cos bx 
(lx 
I o (ft 2 + .r 2 )' ! 
has been evaluated when n is a positive integer,'" the formula in this case being 
cos bx -j _ 7r 1 f, , »(?i — 1) /' 1 \ , {n+l)n(n—l)(n—2)fV\ 2 ( ^ 
Jo (a 2 + x 2 )* Ch: ~~2\n-1)\ 4W| 1 + ~2 [abj^ 2.4 \ab) + 
e~ ab 
(14). 
* See Schlomilch, '‘ Analytiscte Studien’ (Leipzig, 1848), part ii., p. 97, and Crelle’s Journal, 
vol. xxxiii., p. 273, or Catalan, ‘ Liouville’s -Journal,’ ser. 1, yoI. v., p. 110. 
