820 
MR. J. W. L. GLAISHBR ON RICCATI’S 
so that (56) becomes identical with (57) if 
a=y/(—l )=i', p+\= v . 
We may therefore pass from the solutions of the equation (1) to the solutions of 
Bessel’s equation (57) by multiplying by x~ k and putting a= v / (— 1), p=v — 
44. The Bessel’s Function, may be defined for real values of v greater than 
— l by either of the formulae 
x v 
2T(i/+l) 
2(2z/ + 2) 2.4(2y+ 2)(2v + 4) 
■&c. 
(58), 
(59), 
where i denotes, as throughout, — 1). 
Comparing (58) with the expression A" in art. 3, we see that if v=p-\-^, and if a is 
replaced by i\ the series in the two formulae become identical, the exact relation 
between V and Bessel’s Function being 
Y=A.FJ /J+ 1 (if ax ), 
where A denotes the constant 
f) 
and p is supposed to he positive. 
The formula (59) corresponds to Mr. Gaskin’s definite integral solution (38) or to 
one of the definite integrals in Hargreaves solution (55). 
45. It is known that J”(.r) may be exhibited as the sum of two series multiplied 
respectively by sin x and cos x, viz.* 
Jl, ( X )~2 T(I + 1 ) ( A C0S ® in X ).( G0 )> 
where 
2z/ + 3tf 2 (2i/ + 5)(2z, + 7).r 4 (2^ + 7)(2 2 v + 9)(2^ + ll) .i- 6 
x— 2v+2 2! + (2z, + 2)(2i/ + 4) 4! (2i/+2)(2y + 4)(2z/ + 6) 6! + &C- ’ 
p_ _ 2z> + 5 .r 3 (2y + 7)(2i/ + 9) A’ 5 (2y + 9)(2i/ + ll)(2z/+13) x 1 p _ 
) ~" , '~2v + 2 Zl~'~(2v + 2)(2v +4) 5! (2i/ + 2)(2i/ + 4)(2v + 6) 7!~ i ~ " C ‘ 
* Lommel’s ‘ Studien liber die Bessel’schen Functionen ’ (1868), p. 17, or Todhunter’s ‘Treatise on 
Laplace’s Functions, Lame's Functions, and Bessel’s Functions’ (1875), p. 292. 
