388 REPORTS ON THE STATE OF SCIENCE. 



and are evidently the orthogonal functions derived from the symmetrical 

 kernels 



6 



G(s,x) = \K(s,t)i<(x,t)dt 



p. 



H(s,x) = \i<.{z,s)K{z,x)dz 



The roots X„ are all positive since G(s,x), ~K(s,x) are kernels of 

 positive type, and are the same for both G(s,x) and H(s,x). These 

 orthogonal functions <p„(s), \f/ n {t) are used by Picard in finding the con- 

 ditions that the equation 



f{x) = \ K {x,t)f(t)dt 



may be soluble, and many other theorems connected with an integral 

 equation of the first kind may be expressed in terms of them. 



Hilbert's expansion theorem does not, however, apply to the important 

 case in which the function f(x) has singularities or i3 discontinuous. 

 The work of Dixon, Kneser, and Hobson remedies this deficiency. 

 Kneser considers series of the Sturm-Liouville type, and removes the 

 restrictions imposed by Stekloff and Hilbert, that the function be con- 

 tinuous with its first and second derivatives, and satisfy the same boun- 

 dary conditions as the normal functions. 1 Hobson has carried the theory 

 a step further by introducing Lebesgue integrals, and considering the 

 case in which the function has singularities. He obtains a very general 

 convergence theorem and applies it also to series of Legendre and 

 Bessel's functions. Dixon considers the very important class of harmonic 

 expansions of the type 



K(x,t) = 2 a n fjx) \j/ n (t) ; 



his paper contains many interesting results. 



Eecent work on the subject has also been done by Birkhoff 2 and 

 Mercer. 3 



Dixon and Filon discuss the expansion by means of Cauchy's theory 

 of residues ; the work is interesting, and seems to suggest that Cauchy's 

 method may be of value in the treatment of integral equations if we 

 consider contour integrals in which the determinant D(A) appears in the 

 denominator. 



Various other expansion theorems connected with integral equations 

 have been considered by the author, but for these we must refer to the 

 original papers. 



14. Integral Equations of the Fust Kind. 

 An integral equation of the type 



f(x) = L(x,t)f(t)dt . . . . (1) 



1 Mason removes a restriction imposed on the differential equation that one of 

 the functions should be positive. 



» Amer, Tr<ms., 1908, vol. ix., p. 373, * Phil. Trans. A„ 1910-11, 



