104 Trans. Acad. Sci. of St. Louis. 



(17)2 -^1 a;*'[j'„(ax)J_„(ax) + o^-„+i(ax)e/„+j(ax)] l 



(17)3 ^|iK'[e/i„(ax) — J_„+i(ax)J_„_i(aa;)] | 



= 2x«/_„(ax), 



when 92 is not an integral number. 



If, instead of applying formula (4), we had made use of 

 formula (4)'; or, which amounts to the same thing, if we 

 integrate both sides of the relations (12), (13), (16) and 

 (17), we arrive at the formulas: 



(18), x\^aJ„{^x)Jn^,{ax)—^J^{ax)J^^,{^x) | 



= (^a^ yg2^ I xJn((t'X)Jn{^x)dx + CoUSt., 



(18), X I aK,(^x)J„+,(ax)—^Jn(ax)K,^,{^x) | 



= (.»-^)J.^„(..)/.„(,.).. + C„nst.. 



(18)3 X I aK„ (^x)K„+,{ax) — /3Kn (ax)K^^,(^x) | 



= (a2_^2) I x^„(ax)^„(ySx)<Zx + Const., 

 when a ^ /8, and 

 (19), x'^ij^(ax)—J„_,(ax)J„+^(ax)\ 



' (ax)dx + Const. , 



==2\xJl 



