Orr — Extensions of Fourier'' s and the Bessel- Fourier Theorems. 233 



Part II, — Seeies. 



Read Febrl'ary 8. Ordered for Publication Febbuauy 24. Published June 14, 1909. 



Inteoduction. 



In various problems in Mathematical Physics it is required to expand, for 

 values of x between «, &, an arbitrary function of x in the form of a series 

 consisting of sines and cosines, or of conjugate Bessel functions of given order, 

 of \x, where the admissible values of X are determined by the aid of certain 

 conditions to be satisfied by the paired terms of the sum for the values a, &. 



What may be called the ordinary sine or cosine Fourier sum theorems are, 

 of course, particular cases. The forms of the series for the more general case 

 of the type arising in physical investigations are well known. In two of the 

 most interesting cases, one of circular, the other of Bessel functions, the series 

 were given originally, I believe, by Fourier,* without a rigorous proof. Since 

 his time the subject has received attention from many mathematicians. My 

 acquaintance with the literature of the subject is so slight that any reference 

 which I can make will probably be misleading. I may, however, mention 

 Dini,-|-Picard,:j:Dixon,§Filon,|| andCarslaw,^ as having given rigorous investi- 

 gations of various theorems of the type alluded to.** So much has been done 

 in the matter, and so much with which I am unacquainted, that I should find 

 it difficult to express an opinion how far any feature of novelty may be 

 claimed for the present paper. It may be thought, indeed, that the proofs 

 of the two leading theorems which I give are almost obvious from the work 

 of Carslaw. They are, however, in some respects of greater generality than 

 any which I have seen rigorously established ; and I may state that I obtained 

 them to some extent independently of other writers. 



* " Theorie Analytique de Chaleur." 



t " Serie di Fourier." 



% " Traite d' Analyse," n., chap. vi. 



§ "A Class of Expansions in Oscillating Functions," Proc. Lond. Math. Soc, ser. 2, vol. iii. 



II " On the Expansion of Polynomials in Series of Functions," P. L. M. S., ser. 2, vol. iv. 



IF " Fourier Series and Integrals," chap, xviii. 



** The expansion alluded to in the first sentence of Art. 1 and that used in Art. 6, besides being 

 included in Dixon's work, have been deduced from the general theory of integral equations : Kneser, 

 Malh. Ann. 63. 



