OiiR — Extensions of Fourierh and the Bessel- Fourier Theorems. 223 



and the denominator as 

 tt" 



, . , W{\a) + J_n'{\a) - 2 cos utt /„(A«) /_„(Aa) }• (62) 



4sin-W7r \ I >-\ n \ 1 



It should be noted that the numerator is a uniform function of A of even 

 degree, and that the fraction vanishes when A is zero. The equation may also 

 be expressed in either of the forms 



Lt. 



r/( 



r* 



/( = » 1 \d\ 



Kn{-- i\r) 



{Kn{i\p)Kn{- i\a) - Kn{- i\fi)K„(iAa) ]fj(p(p)d,>* 



= -'^{<P0'-^) + <pO' + ^yh (63) 



-,^i-rx.uf'^"(-'^'') 



h J a J^n (- *Ao) 



X [K^iXr) !{-,,(- iXa) - Kn{- iXr) K„{iXa)] p<p(fj)dp* 



= -'^{cl>{r-e)-^<p(r + ,)], (64) 



wherein the path of A is a contour passing above the origin, and the initial 

 arguments of iXr, iXp, iXa are 37r/2 ; the final, 7r/2. The contour may be, and 

 in the first instance is, supposed to pass indefinitely close to the origin ; by 

 so doing, each of the forms {6oj, (64) is seen to be equivalent to that used in 

 (60) ; but the path cannot pass throitgh the origin which is a singularity, 

 (a branch-point, not a pole), of the fractions in (63 j, (64). 



As usual, the range of p is divided into two parts — one from a to r, tlie 

 other from r to ^. 



A point of some subtlety which may arise in identifying (60) with (63), 

 [or (64)], should be noted. 



['AfU^^^ {K„(iXp)Kn{- iXa) - K„(-iXp)K,{iXa)\, 



occurring in the former, is equated to 



" AfZA f4^^ {K„{i^p)K„{- iXa) - K„{- iXp)K„{iXa)\. 



-h -ti.n{- IM') 



* These expressions closely resemble one usud by Carslaw, " Fourier Series and Integrals," 

 p. 393, 1. 15 ; p. 399, 1. 19, but were arrived at without any knowledge of bis Mork in ibis 

 connexion. Although be does not state uor use equations (60), (63), or (64), I apprehend, from the 

 concluding paragraph of his book, that he is aciiuaiuted Avith them, expressed either as here or in 

 some equivalent form. 



In conne.\ion, however, with the discussions in the present paper as to convergence, &c., 

 I constantly consulted this volume : I would acknowledge my indebtedness and express my high 

 appreciation of the book throughout. 



[32*] 



