ON THE THEORY OP INTEGRAL EQUATIONS. 417 



geneous integral equation possesses an in6nite number of linearly 

 independent solutions. 



An example of another nature is provided by the integral 



TT 



r sin (1 - a)(H-ji) du = * ( < a < 1) 

 J sin (i — u) (sin ti)" (sin £) a 



o 



obtained by Hardy. 1 In this case the kernel is infinite, for (S = o,u = r), 

 ($ = 7i-, u = o) and the normal function (cosec i)" is infinite for (£ = o,tt). 

 Integral equations with the kernel (s + t)~ x are of special interest, 

 and have been studied by Stieltjes, 2 Hilbert, and Weyl. This kernel is, 

 of course, derived from e~ $x by forming the iterated function 



c 



" . e~"dx. 

 Stieltjes shows that the equation 



/<*> = f*4 : 



t 



may be solved by means of the formula 



f{t)= I Lt [/( - S -ie)-f(-S+ i:)] 

 We->o 



Hilbert 3 shows that by using the orthogonal system sin pirt we may 

 pass from the kernel to a quadratic form of the type 



S -p t 



CO C£> r» ft* 



2 2 35L. + K*(x) 

 p=l3=1 p + q 



where K*(x) is a limited continuous form. Hilbert 4 shows that the 



form 2 _?p^ is limited, and another proof of this result has been given 



pi p + q 

 by P. Wiener. 5 



Weyl has considered the kernel — r— -. in connection with the 



s -f- t 



orthogonal functions of Laguerre. The kernel is also of importance in 



the theory of differential equations of the hypergeometric type. 



The integral equation associated with a linear differential equation 



becomes singular when the interval which defines the limits of the 



integral contains a singularity of the differential equation. The 



simplest case arises when the differential equation is 



1 Quarterly Journal, 1901, vol. sxxii., p. 384. The paper contains another 

 example of a similar nature. 



2 Annates de Toulouse, 1894, t. 8 ; 1895, t. 9. See also Borel's Leqons svr les scries 

 divergentes. 



8 Lectures, 1906, Summer Semester, 



4 Mathematische Oesellscliaft, 1907; cf. Wejl's dissertation, p. 83. 



6 Math. Ann., 1910, Bd. 68, p. 361. 



• Cf. H. A. Webb, Phil. Trans. A., vol. cciv., pp. 481-497. 



