776 
MR. J. W. L. GLAISHER OX RICCATI’S 
where 
. t (—iy n o n , ..1.3.5 . . . (27-1) 
~ c ~ a (i+ 1 )!^'- 2 ' ‘ * 2 ) — ( )' 2 . 4.6 . . . (. 2 i+ 2 ) a ' 
Of the terms involving even powers of a the first that contains a term in h i+1 is 
x i+1 (x+h) i+ \ 
7 2i+2 
(27 + 2)! 
so that the coefficient of h >+1 in the terms involving’ even powers of a 
v2£+2 
y +1 +77T 
v 2i+4 
-G-L2W +3 -i- o8t+6 — 0 + 2 )0 + 3 ) Z -+5_L^. C 
(27 + 2)f~ 1 (27+4)k + ' + (2i + 6)! ^2! 
v2i+2 
•af +1 -{ 1 
1 c 2 /; 2 
1 
—+ &C. \ — 
722+2 
(27 + 2)! 
(27 + 2)! [ ' i +1 2! 0+1)0 + 1) 2 4 2! 
The complete coefficient of /t :+1 in the expansion of e a ' /{x ~ +xh) therefore 
27 + 1 
y. 
-.2i+2 
y 
XU+ (2i + 2)!^ X j U + ( )' cr ' +1 (l.3.5 . . . 27 + l) ! 
= \{Tj-gV}, 
g being the same as in art 5. 
11 . Now 
g<z *J(x* +xh) — ggg- gg| VC?°++—-rj — g—gr g°j •J^+xh)+x j 
and we obtain other forms of the integral bj finding the coefficients of h l+l in the 
expansion of ++-■*} and of e a { V( ^ +x/l)+T }, and multiplying them by e ax and e~ ax 
respectively. 
It is well known that 
1 _ 1 ”= 2-t-1 1 +nt + +p) <a+ ”0+y( ,, + s p3 +&c . j _ 
and 
ldV(l-4$) l =2” 1-wM 
3! 
n(n—o) n %(%—4) (% — 5) . 
y! 
3! 
+ &c. L 
where in the second series, if n is an even positive integer the coefficients of the 
\n —1 terms involving t hl+1 , A" +2 . . . V‘~ l are zero, and if n is an uneven positive 
integer the coefficients of the — 1) terms involving t ite+1) , id ( " +3) . . . t H ~ l are zero. 
Putting t=—~, these formulae become 
& 4x 
{ <y(x l +xh)—x } n =— h n 
A 
{>/ (x° -\-xli)-\-x}"= 2”x n 
1 —n 
1 + 11 
h ( n(n + o) 
4aP 2~! 
h , n(n — 3) 
4a: 2i~ 
li 3 n(n + 4)(n-j- 5) 
4A7 2 3!~ 
h 3 t n(n—4:)(n—5) 
4hA ' 3!~~ 
