216 



Proceedings of the Royal Irish Academy. 



when n + 1 is positive, each element of the integrand is multiplied by e-^^^^ 

 where t is a positive quantity, which is afterwards diminished indefinitely. 

 And, in ordinary cases, the limit of the integral, when t is indefinitely small, 

 gives its value when t is zero. We have, when n + 1 is positive, 



Hence, 



p-+r- 



p- + r- 



(41) 



-^'-^Jn{Xr)J,,fXp]Xd\ = (l-e--)i-^/„(^|yt (42 



This last equation may be extended to values of n for which n + 1 is 

 negative. By means of the equations 



we obtain 



dn-i{x) = nX-'Jn{of) + J',,{x), 

 Jn+l{o:) = 7lX^Jn{x) - J'„{x), 



c-^'^J„-,{Xp)Jn-^{Xr)Xd\- c-''U,„,{Xp)Jn,,iXr)XdX 



c-'"-'{2np~'J,,{Xp)J\,(Xr) + 2nr-^Jn(Xr)rn(Xp)]dX 



(43) 



e-^'^2np-h-hI{J„{XpyMXr)] 



e-^'^4:np-'r-'XtJn{Xp)Jn{Xr)dX, 



(45) 



on integrating by parts. 



Thus, assuming that (42; holds for n and for [n + 1), we derive the equations 



cr^'^J„^,{Xp)J„-^{Xr)XdX 



o • c-(P'^'-'^^' {4:nt ^ (pr ^ (pr 



= (1-e^""') /„./^ 



2t 



2t 



i.e. the equation (42) holds for n - \. And so by induction all restrictions 

 o]i the value of n may be removed. 



* Weber, Crelle's Jounial, Ixis. ; Hankd, Matli. Ann., yiii., s. 470. Graj- and Mathe'vrs ; 

 " Besuel Functions," p. 78. 



t Macdonald, Proc. Lond. Math. Soc, xxxv., p. 438. 



