OuK — Extensions of Fourier'' s and the Bessel- Fourier Theorems. 213 



where A, B, C, D are some numbers independent of p and of A ; that is, when 

 1 Xp I is sufficiently great ; and the indices of the exponential terms have large 

 negative real parts except at the limits of integration. 



Interchanging the order of integration, the integral in X arising from the 

 first term is of order /r^ when p and r are vmequal, and finite when they are 

 equal ; thus the first term tends to the limit zero. The integral in X arising 

 from the second term is of order h"^ for all values of p ; thus the second term 

 has zero for limit also. 



Consequently, equation (16) is established. 



Art. 4. The equation 



Lt f* f 



h = X \Kn {- i\r) dX K„ (- i\p)p({> (p) dp = 0. 



cj-h J a 



In a precisely similar manner we may establish the equation 

 Lt. ^^ 



-Lt I I 



A=» XdX Kn (- iXr) Kn{- iXp)p(f)(p)dp, 



cj-h J a 



or, 



4 sin-?i7r 



XdX 



{^"-j:„(Ap)j:,.(Ar) + e-"-^J„{Xp)Jn(Xr)]pcj>{p)dp = 0. (27) 



For, if we use the asymptotic values of the K functions, the left-hand member 

 is replaced by 



iTT Lt. 



IX\ e^^i'>''')(plr)mp)dp, 



(28) 



which is zero, also by Fourier's integral theorem ; and the same reasoning as 

 above shows that the difference between (28) and the left-hand member of (27) 

 is zero. 



Art. 5. The equation 



XdX j' J„(Xr)J„(Xp)pcp{p)d<p = i (1 - e--)'/' (^ - 0- 

 Combining (16) and (27), we obtain the two results — 



XdX 



XdA 



J-„ (Ar) /„ [Xp] p4> (p) dp^iil- r'"^') <l> {r - e\ (29) 



J^{Xr)J^n{Xp)pHp)^P = Kl - e-'"^^)<p{r - .), (30) 



r > a. Of course, each is zero if r = a. 



E.I. A. PROC, VOL. XXVII., SECT. A. 



[311 



