ON THE THEORY OF INTEGRAL EQUATIONS. 365 



developed by C. Neumann, Poincare,' Picard 2 and Korn, and applied to 

 problems in elasticity by E. F. Cosserat, 3 Korn, 4 and others. A good 

 account of the method is given in E. E. Neumann's prize memoir 

 (Leipzig, 1905), and in Korn's ' Abhandlungen zur Potential Theorie ' 

 (Berlin, 1901). Neumann's method solves the problem of Dirichlet for 

 the case of a convex region, and the solution may be extended to regions 

 of a more general character by Schwarz's method of alternation. Further 

 researches in this direction have been made by A. C. Dixon. 5 



Other methods of solving Dirichlet's problem given by Kirchhoff, 1 

 Eobin, 7 and Stekloff, 8 are closely allied to Neumann's method. 

 Poincare 9 has devised a method in which a series of functions is 

 formed, each of which satisfies the boundary condition, and whose 

 limit satisfies the differential equation as well. Hilbert 10 has shown 

 that the old method of the calculus of variations can be remodelled 

 and made quite rigorous, thus realising the hope expressed by Brill and 

 Noether that the principle of Dirichlet would some day be revived. 

 Generalisations of Dirichlet's problem to partial differential equation of 

 elliptic type have been given by Segre Bernstein. 11 



The advent of Fredholm's method of solving the integral equation of 

 the second kind has provided a new and beautiful method of solving 

 Dirichlet's problem, and has led to many new developments. It is true 

 that the method does not apply to the general case. For instance, 

 Kellogg remarks that if the boundary is a rectangle, Fredholm's series 

 does not converge. It is probable, however, that by some modification 

 of the method this difficulty may be overcome. 



Neumann's solution of the equation 



b 



/(s) = ^(s) - \L(s,t) <p{t) dt 



a 



may be expressed in a concise form by introducing the iterated functions 



b 



K 2 {s,t) = \K(s,x)^(x,t)dx, 



a 

 b 



K 3 (s,t) = h(s,x)K. 2 (x,t)dx, 



a 



b 



K n (s,t) = *.■(«,&)»:„_,(#,<)<&, 



a 



1 Acta Math., 1897 ; Rend. Palermo, 1894. 



2 Traite d'Analyte, 2nd Ed., vol. i. * 3 Comptes rendus, 1. 133 (1901). 

 4 Ann. de Toulouse, 1908 ; Ann. de I'Ecole Normale Supirieure, 1908. 



s Cambr. Phil. Trans., vol. xix., part 2, p. 203; Proc. London Math. Soc, ser. 2, 

 vol. i., p. 415 (1903). 



6 Acta Math., 14, 1890, p. 180. » Comptes rendus, 104, 1887, p. 1834. 



8 Ibid., 125, 1897, p. 1026. 



9 Theorie du potential Kervtonicn, Paris (1899), p. 260. 



10 Math. Ann., Bd. 59, 1901. 



11 Comptes rendus, Nov. 1903; Math. Ann., Bd. 62, 1904; Bd. 69, 1910; recent 

 papers on Dirichlet's problem have been published by Fubini, Rend. Palermo (1907), 

 and Lebesgue, ibid. (1909). 



1'JlU. B B 



