106 Trans. Acad. Set. of St. Louis. 



(21)2 x^ij„ (ax)J^(ax) + J_^_^^{ax)J^+,(ax) | 



= 21 xJn (ax)J_n{ax)dx + Const., 

 (21)3 x'^ij\{ax) — J_,,_,{ax)J_„+^(ax) I 



r 2 



= 21 xJ _n (ax )dx-\- Const. 



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

 (21)j, 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 /S^a and n being an integ- 

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



(22) aJ„(^)J,_,^{a) — ^J„{a)J,^^(^) 



= (a2 — ^) \ xJn{a,x)Jn{^x)dx^ 





(23) aK,(^)J„^^(a) — fiJ„(a)ir„^^{^) 



= (a2_/32) I xJ„(ax)K„{^x)dx, 





(24) e7^(a)— e7'„_Xa)J'„+,(a) =2 j xJ^{ax)dx, 



(25) Jn{a)KnW-Jn-MK„^^(a) 



i I xJ„i 



2 I xJn(ax)K„(ax)dx. 

 



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

 (26) aJ_„(^)J,_^(a) + ^J„(a)J_r,^,(^) 



a'O j xJ„i 



= (^ — a''') I xJn(ax)J_n(^x)dx, 

 



