274 Proceedings of Royal Society of Edinburgh. [sess. 
and an indicial function 
/-* [* [ ” 1 - [„ 
^[a+2] 
n + 2] [a + n + 2] 
^[a+4] 
[a - n + 2] [a + n + 2] [a - n + 4] [a + n + 4] - 
The principal roots of the indicial equation are 
.] 
a= + n , a = —n . 
If n he not an integer, the corresponding integrals are and 
J[-n] ' 
1 j a*[w+2] ^ 
J [»1 (*) = [2] n r p2 ([n + 1]) r W " [2] [2w+ 2] + • f 
1 | ^[-tl+2] ! 
J[-»](*)= [2]-»IX[l -»]) \ [2] [2 - 2 n] + ■■ } 
If n = 0 these integrals are identical, while if n be an integer, 
one or other becomes ineffective according as n is positive or 
negative. In these cases, then, it is necessary to form a second 
distinct and effective integral corresponding to HankePs solution 
of Bessel’s equation. 
When a is integral, we write 1 
/(§C{ [a+2n]-[n] J- | [a + 2n] - [ - n] | 
[a] _ ^[a+2] 
x [a - n + 2] [a + n + 2] + . . . 
... - ( - 1)’ 
— 2 ] 
] 
+ E 
Sj[a + 3r]-[»]} )[« + 2r]-t-*]} 
2 71 -j- 2] 
x [a+ 2 n ] _ | [ a + 2n + 2] - [n\ | j [a + 2n + 2] - [ - n\ j 
= oq + o> 2 . 
1 Cf. Forsyth, Theory of Differential Equations , vol. iii. pp. 101, 102. 
