822 
MR. J. W. L. GLAISHER OX RICCATI’S 
Putting v=p-\-\, we liave therefore 
•J /J+ '(,r) = 
- i ,p+1 ( 
\/ (27r.r) 
where 
2?0 + l) (p—l)p(p + l){'p + 2) 1 
2 2.4 (i’x 
r 
and /3 l denotes a similar series, having all the terms positive. 
If p is a positive integer, this expression corresponds to y(IF — P'), when the terms 
in IP and P' are written in the reverse order, as in § IV. If p is not an integer, the 
series, as already mentioned, are divergent, so that, strictly speaking, the formula only 
has a meaning when it contains a finite number of terms. An expression can however 
be found for the remainder after a finite number of terms, i.e., for the difference between 
J p+i (x) and the sum of these terms, by means of which the use of the formula in 
calculation may be justified. 
47. It is a known theorem in Bessel’s Functions that if p is a positive integer, .P(x) 
is equal to the coefficient of z p in the expansion of e 2 ' z ’ ; and it follows therefore, by 
means of the relation between Y and J p+i (i'ax) in art. 44, that, if p)-f-I= an even 
positive integer =2 to, 
, T , . 2 2fl T(2m + l) i p,-. . , „ „ . ax[ 1 
\ =( —- x ! X coefficient of :~ w m cos— (z — 
and if p-\-±= an uneven positive integer = 2m 1, 
Y=(-)' 
2 2«.+i r (2m + 2) , 
X coefficient of z~ m+1 in sin 
ax 
48. It was shown in § II. that the differential equation (1) was satisfied by the 
coefficient of h p+l in the expansion of e "' / g+- 7 ') j anc [ we thus find that if v=p-\-\, p 
being a positive integer, the general integral of Bessel’s equation (57) is 
w—x *{ c 1 X coefficient of in the expansion of cos x /{x z -\-xli) 
>5 
sin *y(x~-\-xh)}. 
5 ? 
>5 
