Chessin — On Some Relations Between Bessel Functions. 105 

 (19)2 x^\jn{ax)Kn(ax)—J„_^{ax)K„^X^x) i 



=: 2 I xJ„ (^ax)E'n (a'X)dx + Const., 



(19)3 x^ I Kl(ax) - K„_,(ax)K„^,(ax) | 



r 2 



= 21 xK„(ax)dx + Const., 



when /3 = a. Both sets of formulas (18) and (19) refer 

 to the case of an integral n. When n is not an integral 

 number these formulas should be replaced by the following 

 ones: 



(20), x^aJ„(^x)J„^,(ax) — ^J„(ax)J„^,(^x) | 



= (a^ — y8^) I xJn {ax)Jn (^x) dx + Const., 



(20), X I aJ_„(^x)e/„_/aa;) + ^J^ {ax)J_,^, (^x) } 



= (^_a2) \xJ„ (ax) J^(^x)dx+ Const., 



(20)3 X I aJ_,(^x)J.,^^{ax) — ^J_r.{ax)J_„^,{^x) | 



= (a2_ y82) I xJ_„(ax)J_^(^x)dx -h Const., 



provided a p£ /3. If /3 = a, the last relations assume the form 

 (21), x2|j„'(ax) — J-„_,(ax)J-„+X«^)| 



r 2 



= 21 Xe/«(aa;)c?ccH- Const., 



