ON THE THEORY OF INTEOKAL EQUATIONS. 370 



* 



i at the point \ = X„ and this root is of multiplicity p, then we have 

 the inequalities 



n <p — {i — l) <p 



i>l 



The solving function is thus certainly infinite for X = X„. 



For further properties of the determinants and the solutions of the 

 homogeneous equations, the reader is referred to the papers of Fredholm, 

 Plemelj, Bryon Hey wood, Goursat, Mercer and Dixon, and the following 

 books, Kowalewski's ' Determinants,' Horn's ' Partial Differential 

 Equations.' 



Fredholm has invented an ingenious device by means of which a 

 system of integral equations 



/,(*) = <j>,(s) - X I K r ,js,t)$J$dt (r = 1, 2 . . n) 



m = 1 



may be replaced by a single integral equation. 

 We define 



/(s)=/,(s-r + l) r-l5s<r 



<t,(t) = <j> r {t - r + 1) r - 1 < « < r 



*M = "«.(« - r + 1, * - » + 1) ^ 1 1 < V< m 



The system of equations may then be written in the form 



b 



f(s) =</,(s) -xL{s,t)f(t)dt. 



a 



A similar artifice may be applied to a system of integral equations of the 

 first kind. 



The theory of systems of integral equations and equations in which 

 there is more than one unknown function has been developed by 

 Cauchy, 1 Murphy, 2 Volterra, 3 A. C. Dixon, 4 Bounitzky, 5 and Cotton.' ; 

 The latter considers a system of integral equations in connection with a 

 system of linear differential equations of the first order ; while Bounitzky 

 develops the theory of Green's functions for systems of differential 

 equations. 



Fredholm's method can easily be extended to integral equations 

 involving surface and volume integrals. It is convenient to adopt 

 Plemelj 's notation and write f{p) for a function of the parameters 

 specifying the position of a point P. The whole of the present work then 

 applies without much modification. It is, however, important to notice 

 that in two dimensional problems, kernels which become infinite, like 



lo,' 



are certainly admissible as the iterated kernel is bounded. Also in three 

 dimensional problems a kernel which becomes infinite like 



1 Mcmoire sur la thcorie des Ondcs, 1815. 



3 Cambr. Phil. Trans., 1833, vols. iv. and v. ' Annali dl Matcmatici, 1897. 



1 Ibid., vol. xix., part 2, p. 203. 



5 Darboux's Bull., 1907 (2), t, 31 ; Liouville's Journal, 1909, t. 3. 



• Hull, de la Soc. Math, de France, 1910, t. 38, fasc. 1-2. 



