ON THE THEORY OP INTEGRAL EQUATIONS. 3G9 



Sommerfeld in the ' Encyklopadie der Mathematischen Wissenschaften ' 

 and Mason's lectures at the Newhaven Mathematical Colloquium. 



The problem of the determination of periodic solutions of differential 

 equations is also reducible to the solution of an integral equation and 

 has been discussed by Mason, 1 Lalesco, 2 A. Myller, 3 and others. 



9. Fredholm's Method. 



The beautiful researches of Fredholm 4 and Hilbert 5 on the integral 

 equation of the second kind have opened out a wide field of research 

 which now occupies the attention of a great number of mathematicians. 

 The Paris Academy recognised the value of Fredholm's contributions to 

 science by awarding him the Poncelet prize in 1908. 



In Fredholm's method the integral equation 



i.(s,t) = K(s,t) — \h(s,x)K(x,t)dx 



a 



satisfied by the solving function is treated as the limit of a system of 

 linear equations. 6 The ordinary solution of such a system as the ratio 

 of two determinants now takes the form 7 



where D(\ ; s,t), D(A) are the integral functions of \ defined by the 

 equations 



b b b 



D(X ; s,t) = ,(s,t) - £-j r({ j)*,, + * J f ,({ J J*)*, dx, . . . 



a a a, 



dw = i -fj.g;)*. + 1 Hit', S) & ' * 



" 2 ' ' ") is used to denote the determinant 



2/1. 2/2 • • y,J 



K{x u y{) *>(z 2 >2/i) . . K (x n , y{) 



<f(«ll 2/a) r (*ll Vi) ' • "( X n, 1/2) 

 KiPuVn) K&n>y») 



1 Liouville's Journal, 1904 ; Trans. Amer. Math. Soc, 6, 1905, p. 159. 



2 Comptet rendus, 1907. 3 Ibid., 1907, p. 790. 



4 C. R. Stockholm, Academy, 1900, vol. lvii. p. 39; Comptes rendvs (Paris), 1902, 

 pp. 219, 1561 ; Acta Math., vol. xxvii., 1903. Fredholm has recently issued a short 

 report on the development and present state of the theory of integral equations 

 (Compte rendu du Congret des Mathcmativiens tenu a Stockholm, 1909). I have not 

 yet had the opportunity of consulting this. 



4 Gbtt. Nachrichten, 1904, Hefte i.-iii. ; 1906, Hefte ii.-v. The method was 

 described by Hilbert in Seminar and in lectures at Gottingen, 1900-01. His work 

 is quite independent of that of Fredholm. 



6 This method of treatment is carried out in detail by Hilbert. See also Plemelj, 

 Monatshefte fiir Math., 1904. The fundamental idea was also used by Volterra, 

 Betid. Lined, 1884. 



' The relation between this solution and the one obtained by Neumann's method 

 is discussed by Kellogg (Gott. Nachr., 1902), Plemelj and Poincare (Acta Math.., 

 1909). 



