Mat 26, 1911] 



SCIENCE 



805 



solution of certain differential equations 

 whose integrals are subject to certain con- 

 ditions as to the limits, and this is a very 

 important problem for mathematical phys- 

 ics. Fredholm solved it in certain partic- 

 ular cases and Picard generalized his meth- 

 ods. Hilbert made a systematic study of 

 the question. 



Consider an integral equation 



where u is an unknown function of one or 

 several variables, / a known function and 

 A any linear differential expression. This 

 eqiiation with the same right as an integral 

 equation may be considered as an infinite 

 system of linear equations connecting a 

 continuous infinity of variables, as a sort of 

 linear transformation of infinite order, en- 

 abling us to pass from / to u. If we solve 

 this equation, we find 



8{f) this time presenting itself under the 

 form of an integral expression. 



Then A and 8 are the symbols of two 

 linear transformations of infinite order in- 

 verse one to the other. The kernel of this 

 integral expression 8{f) is what we call a 

 Green's function. This function was first 

 met in Dirichlet's problem, then it was 

 Green's function properly so called, too 

 familiar to be stressed ; we had already ob- 

 tained different generalizations of it. To 

 have given a complete theory belongs to 

 Hilbert. To each differential expression 

 A, supposed of the second order and of 

 elliptic type, to each system of conditions 

 as to the limits, corresponds a Green's 

 function. We cite the formation of the 

 Green's functions in the case where we 

 have only one independent variable and 

 where they present themselves under a par- 

 ticularly simple form, and the discussion 

 of the different forms the conditions as to 

 the limits may assume. That settled, sup- 



pose we have solved the problem in the case 

 of an auxiliary differential equation differ- 

 ing little from that proposed and anyhow 

 not differing from it by the terms of the 

 second order; we can then by a simple 

 transformation reduce the problem to the 

 solution of a Fredholm equation where the 

 role of kernel is played by a Green's func- 

 tion relative to the auxiliary differential 

 equation. However, the consideration of 

 this auxiliary equation, the necessity of 

 choosing it and solving it being capable of 

 still constituting an embarrassment, in his 

 sixth communication Hilbert frees himself 

 from it. The differential equation is trans- 

 formed into a Fredholm equation where 

 the role of kernel is played by a function 

 our author calls parametrix. It is subject 

 to all the conditions defining Green's func- 

 tion, one alone excepted, the most trouble- 

 some, it is true; it is not constrained to 

 satisfy a differential equation; it remains 

 therefore in a very large measure arbi- 

 trary. The transformation undergone by 

 the differential equation is comparable to 

 that experienced by a system of linear equa- 

 tions if we replace the primitive variables 

 by linear combinations of these variables 

 suitably chosen. The method is nowise re- 

 stricted to the case where the differential 

 equation considered is adjoiat to itself. 



Hilbert examined in passing a host of 

 questions relative to integral equations and 

 showed the possibility of their application 

 in domains the most varied. For example, 

 he extended the method to the ease of a 

 system of two equations of partial deriva- 

 tives of the first order of the elliptic type, 

 to polar integral equations, that is to say, 

 where the coefScient a in the integral equa- 

 tion (1) in place of being always equal to 

 1 is a function of x and in particular is 

 equal now to + 1, now to — 1. 



He has applied the method to the prob- 

 lem of Riemann for the formation of fune- 



