220 Proceedings of the Royal Irish Academy. 



The case of n zero requires special consideration, since the asymptotic 

 values are then unserviceable. 



Since J^iO) is unity, the integral is now 



\d\ 



doMp(pip)dp. 



By integrating the J series term by term, we obtain 



XdX 



J,{Xp)pdp^l-J,(y:) = l. (54) 



Hence, using the mean- value theorem, as in the proof of Fourier's theorem, or 

 quoting Jordan's* or Du Bois-Eeymond'sf extension of Dirichlet's or Fourier's 

 integral theorem, it readily results that the value of (40) is (p{e). (If the 

 lower limit were a, different from zero, the value of Hankel's integral for 

 /■ = a would be only 1 {a + e).) 



When a, r are each zero, the integral in (37) is of course infinite, since 

 Yo (0) is infinite. 



It has been supposed, also, that h is finite. A little consideration 



shows, however, that if (p (r) dr is convergent, and (p satisfies Dirichlet's 

 conditions, we may replace 5 by co in the integrals 



Lt. 



A fKn{iXr) ) 



rb 



Xd\ ] Kn{- iXr) I Kn {- iXp) pf (p) dp, 



of which the first two are used in Art. 6, and the last two in this and Art. 7. 

 And thus, when we can, as in Art. 7, convert either or both of the last two 

 into line-integrals in A, h may be replaced by oo in such line-integral or inte- 

 grals; accordingly, so long as Hankel's original form is valid, i.e. when n is 

 positive, or negative and numerically < 1, h may be made infinite. But it is 

 not allowable, without restriction on ^, to use an argument which involves 

 cancelling the products JnJ-n, unless b is finite ; thus, it is not legitimate 

 (nor intelligible) to make h infinite in the contour-integral (14). 



Art. 11. Nature of the Convergence of the Fourier- Bessel Integral. 



It is easily shown that the nature of the convergence of the integral 

 in (14) and the order of magnitude of its coefficients are the same as those of 

 Fourier's.+ For, when A is large, it is evidently legitimate to substitute the 



'^•'"Traited'Analyse,"ii.,p.216 ; lie ascribes it to Du Bois-Reyiiiond. t " Crelle," Ixix., s. 82. 



X Necessarily I do not go into these matters fully. For the discussion of similar questions in 

 connexion with Fourier series (and, to a certain extent, integrals) and the conclusions arrived at, 

 see Carslaw, l. c. below ; "Whittaker, ' ' Modern Analysis " ; Hobson, " Functions of a Real Variable " ; 

 Stokes, "On the Critical Values of the Sums of Periodic Series," Camb. Phil. Trans., viii. ; 

 Mathematical and Physical Papers, vol. i. 



