102 Trans. Acad. ScL of St. Louis. 



= (a2_/32)xJ,(ax>/_„(/3x), 



= (a2 — /32>J_„(aa3)J_„(/3x), 



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

 Now, we know that, whatever be n, 



(9) -d^ - x'^n{^'^)—^Jn+,(^^)^ 



dJJax) ^ r / X I T / \ 



(10) 2; = — ^^«(««^) +«^»-i("^)' 



dKJax) _n 



(11) ^ =x^^'^^''''^~''^"+>^''^^' 



and that similar relations exist when a is replaced by /3. 

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

 pressions for the derivatives of Bessel functions we readily 

 obtain the formulas : 



(12), ^{x[aJ„(/3x)J-„+,(ax)-/3,/„(ax)^„+i(^x)]j 



(12), ^|x[a/i'„(/3x)./„+i(ax)-/9./„(ax)/r,+i(/3a^)] j 



= (a2— /32)xJ-„(ax)/r„(^aj) 



(12)3 -^|x[aA'-„(^x)/iVi(«^)— ^^'n(«a^)/C+i(/3a^) ] } 



= (a2 — /32)x/i;,(ax)/r„(^x) 



when n is an integral number, and 

 d 



(13), ^|x[aJ'„(/3x)J",+i(ax) — /3J„(ax)J„+,(^x) ] 



= (^a'' — ^'i)xJ^{ax)Jn{^x), 



