Chessin — On Some Relations Between Bessel Functions. 107 

 (27) ^„(aK_„(a) + e7_„+i(a)J-„^^(a) 



= 2 I xJn {ax)J_n{ax)dx. 



xJn 



Further, if n denote a positive, but not integral, number, 

 formulas (20)^ and (21)^ yield the relations 



(28) aJ„(^)J„^^{a)-^J,(a)J,^^(^) 



= (a2_^2) xJ„(ax)J„{^x)dx, 



xJ„ 



r' > 



a) = 2 I xJ„ 



(29) Jn{a)—Jn-MJn+rW = 2\ xJ:(ax)dx, 



in form identical with (22) and (24) ; while n being a nega- 

 tive, but not integral, number, we derive from (20)3 and 

 (21)3 the formulas 



(30) aJ_„(^),/_„+,(a)_/3J_„(a)J_„^,(^) 





a)= 2 I xJ_„\ 



(31) e/_„(a) — J"_„_i(a) J_„^,(a)=2 I xJ_^{ax)dx, 



which are, practically, identical with (28) and (29), since 

 — n in (30) and (31) is a positive, but not integral, number. 



The integration between the limits and 1 in formulas 

 (20), and (21), when w<0, or in (20)3 and (2I3) when n>0, 

 is only possible if | n | <1. Hence, formulas (28) and (29) 

 are still valid if — l<w<0; and formulas (30) and (31) hold 

 good also when 0<n<l. 



Finally, in the relations (18)3 and (19)3 the integration 

 may be taken between the limits and 1 when?i = 0. We 

 then obtain the formulas 



