102 Trans. Acad. Sci. of St. Louis. 



of which the first one is identical with (6)^ in form. 

 Now, we know that, whatever be n, 



dJ^ (ax) n 



dJJax) n 



(10) -^^ =—^J^(ax)+aJ^_,{ax), 



dKJ^ax) n 



(^^^ dx ='x^^n{^^) — o^K^^^{ax), 



and that similar relations exist when a is replaced by yS. 

 Substituting into the identities (6) and (8) the above ex- 

 pressions for the derivatives of Bessel functions we readily 

 obtain the formulas : 



= {a-^ — ^)xJ,{ax)J^{^x) 

 (12), '^^x[_aK^{^x)J„^,{ax)—^J^{ax)K,^,{^x)'\ } 



(^^)3 Yx { ^{^K,{^^)K^^,{ax)-mn{cLx)K^^,{^x) ] I 



= (a2 _ ^)xK,{ax)K,{^x) 



when 71 is an integral number, and 



(13), -^^x[aJ,{^x)J,^,{ax) — ^J,{ax)J,^,{^x) ] | 



= (a^ — l3')xJ^(ax)J,(fix), 



