106 Trans. Acad. Sci. of St. Louis. 



(21), x^ij^(ax)J_n(<^x) +J_„+,(a.'»)'/„+i(aa;) I 



= 2 I xJn {ax)J_n{cix)dx + Const., 



(21)3 X^ijLn{ax) — J^n-,i'^Xy-n+,(^^) | 



f 2 



= 21 xJ_n{ctx)dx-^ Const. 



With the exception of formulas (18)3, (19)3, (20)^, (20)3, 

 (21)i, and (21)3, the integration may be assumed between the 

 limits and 1, the only restriction on the number n being 

 that specified above. Thus, with ^^a and n being an integ- 

 ral number, positive or negative (zero, of course, included), 



(22) aj-„(^)e4+,(a) — /3e7„(a)J-„^^(^) 



= (a2_/32)l xJn(ax)Jn(^x)dx, 



(23) aR\{l3)J^+^{a) — ^J,{a)lL,^^(/3) 



= (a2 — jS^^ I xJn{ax)Kn(^x)dx, 



(24) Jn{a) —Jn-MJn+M = 2 1 xjl{ax)dx, 



(25) J„(a)/r„(a) — J-„_,(<i)A'„+,(<i) 



r' 



= 2 1 xJn(ax)Iin{^x)dx. 



Likewise, with /3^a, but n not an integral number, 



(26) aJ_„(/3)J-„_,(a) +/3J„(a)J_„^i(/3) 



= (/32_a''') I xJn(ax)J_n(^x)dx, 



^0 



