ON THE THEORY OF INTEGRAL EQUATIONS. ,389 



in which the limits of integration are fixed has been called by Hilbert l 

 an integral equation of the first kind. The term ' integral equation ' is 

 due to P. Du Bois Reymond. 2 



The function K(x,t) is called the kernel {hern, Noyau), but the terms 

 nucleus, characteristic function are sometimes used. 



Particular forms of equation (1) were considered by Laplace, Fourier, 

 and Abel, but a general theory was first given by Murphy, 3 who con- 

 sidered the equation in connection with electrostatical problems. It 

 should be mentioned that some of the problems considered in Green's 

 essay (Nottingham 1828) are equivalent to integral equations of the 

 firgt kind. 



It was pointed out by Peacock ' that the homogeneous equation 



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



may possess an infinite number of solutions, and it was observed by 

 Murphy and Liouville that the general solution of (1) may be expressed 

 as the sum of a complementary function and a particular integral. 

 The integral equation (1) has been studied by Pincherle 5 in connection 

 with the theory of distributive operations and by Pincherle, 6 Levi Civita, 7 

 and the author in connection with linear differential equations. 



A useful method of solving the equation is to expand the function 

 ^(x,t) in a series of the form 



K(x,t) = 2a n ih n (x)x n (t) 



and the function f(x)<j>(t) in the form 



f(x) = 2c A(z) .... 

 <f,(t) = U n f n (t) 



where the functions f n (t),x„{t) are connected by the relations 



b 



[<Pm{t)Xn(t)dt = 0. m ± n. 



J = 1. m - n. 



This method, which is virtually given by Murphy, has been developed 

 in other forms by Dini, 8 Lauricella, 9 Picard, 10 and the author. 11 Picard 

 has made use of a theorem due to Eiesz on expansions in series of 

 orthogonal functions to obtain an expression for the necessary and 

 sufficient conditions that a function j\x) may be capable of being 

 expressed in the form (1). The condition is that a certain series of 

 constants derived from the function f(x) should converge. lb is 

 necessary of course to impose a number of restrictions upon the 

 function k(x,t). 



1 GUtinger Nachrichten, 1904. » Crelle, 1888, vol. ciii , V- 228, 



3 Cambr. Phil. Trans., 1833, vol. v., pp. U3-148, 315-394. 



* Brit. Assoc, Beport, 1833. 5 Acta Math., 1887. 



6 Acta Math., 1892 ; Chicago Congress. 1892. ' Lomb. Rend., 18S5 Torino At ti, 1 H^f,. 



8 Ami. Univ. Tote, 1880, vol. xvli. Dini's method really applies to an equation 

 of Vol terra's type. 



9 Bend. Lincei, 1908, p. 193. ,0 Bend. Palermo, 1910. 



•' Prop. £ondon Math. Soc, 1907, ser, 2, yol. iv., p. 461 ; Mess. Math., 1910, p. 129, 



