Chessin — On Some Relations Between Bessel Functions. 101 

 n being an integral number, then the function v will satisfy 

 the equation (3) when/(a3) =a^ — — j-^ — , while the function 



w will be a solution of the differential equation (3)' when 



4?i2 — 1 

 cfy(x) = /32 ______ Therefore, by (4), 



(6x ^{K^«(-)-^-^«(^-)-^:)} 



Likewise, n still being an integral number and the func- 

 tions y*(x) and 4>{x^ being selected as above, if we take, in 

 succession, 



(5)2 V = VxJn{ax)^ w = VxKn{^x)\ 



(5)3 V =. V xK^^ax) ^ w ^ V xKn{fix) ; 



we obtain the formulas 



= (a^ — ^)xJ„(ax)K,{^x), 



= (a^ — /3^)xK^(ax)KJ^^x). 

 When n is not an integral number we take successively 

 (7), v= VxJ^{ax), w = \/xJ„{^x) ; 



(7), v = VxJj^ax), w = l/xJ_„(^x) ; 



(7)3 V = VxJ_,,(ax), w = VxJ_n{^x); 



and arrive at the identities 



= {a^ — ^)xJ,(^ax)J,{^x), 



