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



(17), ^lx^[^jl(ax) — J,_r{ax)J,+,(ax)']'j = 2xJl{ax), 



= 2xJn{ax)J_Jax) 



(17)3 ^|x2[j!„(ax) — J_„+i(ax)J_„_i(ax)] j 



= 2xJLn{ax), 



when n is not an integral number. 



If, instead of applying formula (4), we liad 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), a;jaJ„(^x)J"„+i(ax) — /3J'„(ax)J„+j(/3x) 



= (a'-J— /32) I xJn{ax)Jn(l3x)dx + Const. 



(18), x\ aKn(0x)Jr,+,{ax)—^Jn(^x)Kri+i{^x) 



= (a2_^2>) 1 xJn(ax)lLn{i3x)dx + CoUSt., 



(18)3 xl aKn(^x)K^+,(ax) — ^Kn{<^x)J^n+i(^^) 



= (a2_ ^^) 1 xKn {o,x)Kn (/3a5) dx + Const. , 

 when a ^ /3, and 

 (19), x2|j^(ax)— J„_,(ax)J„+,(ax)| 



= 2 I xJ^(ax)cZx + Const., 



