J* 



CO 



ON THE THEORY OF INTEGRAL EQUATIONS. 351 



CO 



o 



has a value A which is neither zero nor infinite, then 



CO CO 



m~\WUi»te--*)\fmi ... (i) 



o — CO 



Hamilton gave as an example the case when (f>(z) is the Bessel's 

 function, 



and introduced the idea of a fluctuating function. The general formula 

 (1) has been established under certain limitations by Du Bois Beymond l 

 and in a more general manner by Dr. Hobson 2 in a profound memoir 

 which contains proofs of many important theorems. 



In 18G2 C. Neumann gave a formula involving Bessel's functions 

 which is a generalisation of Fourier's theorem, and some years later 

 Hankel 3 obtained the beautiful formulae — 



CO 



f(x) = h v (icl)t<t>(t)dt 



o 



CO 



(j,{t) = U v (xt)x/(x)dx 



which contain Fourier's formulae as a particular case. These formulae 

 were discovered independently by Sonine. 



Froofs of Hankel's formulae have been given by Du Bois Beymond,- 4 

 Basset, 5 Nielsen, 6 and Orr." The formulae have been discussed with 

 the aid of convergence factors by Sommerfeld 8 and Hardy. 9 Weyl 10 

 has established them by a general method appropriate for dealing with 

 singular integral equations. 



A remarkable extension of Fourier's formulae has been obtained by 

 Heaviside in his electrical researches. 1 ' If b is defined in terms of u by 

 means of the equation 



tan b = f(u) 



where <■/»(«) is a given odd function of u, then an arbitrary function 



1 Crelle's Journal, Bd. 69, 1868 ; Bd. 79, 1875. 

 - Proc. Lond. Math. Sue, 1909, ser. 2, vol. vii. 

 ' Math. Ann., Ed. 8, 1875, p. 471. 

 4 Proc. Lond. Math. Soc, 1909, ser. 2, vol. vii. 

 b A Treatise on Hydrodynamics, 1888, vol. ii. 



6 Handbuch der Cylinderfunlttionen, 1904 (Leipzig). 



7 Proc. Boy. Irish Acad., 1909. 



8 Dissertation Ebnigsberg, 1891. 



" Cambr. Phil. Trans., vol. xxi., p. 44. 



lu Dissertation Gottingen, 1908. 



" Electrical Papers, vol. i., pp. 122, 158. 



A A 2 



