802 



SCIENCE 



[N. S. Vol. XXXIII. No. 856 



But we can not enter iipon this subject 

 without paying homage to the immense serv- 

 ice which Fredholm has rendered to sci- 

 ence. 



The theory of Fredholm is a generaliza- 

 tion of the elementary properties of linear 

 equations and determinants. This gen- 

 eralization may be followed up in two 

 different ways : on the one hand, by consid- 

 ering a discrete infinity of variables con- 

 nected by an infinity of linear equations, 

 which leads to determinants of infinite 

 order ; on the other hand, by considering an 

 unknown function <f>{x) (that is to say in 

 last analysis a continuous infinity of un- 

 knowns) and seeking to determine it by the 

 aid of equations where this function fig- 

 ures in integrals under the sign f. Upon 

 this second way Fredholm has embarked. 



Let K(x, y) be a function we call the 

 kernel; the integral 



f{x)—yK{x, y)ct,{y)dy, 



taken between fixed limits, may be re- 

 garded as a transform of 4>[x) by a sort 

 of linear transformation and be repre- 

 sented \)j S<^{x). 



The integral equations may then be put 

 under the form 



(1) a4>{x)+\S4,{x)=f{x), 



where f{x) is a given function; the equa- 

 tion is said to be of the first kind if the 

 coefficient a is null, and of the second kind 

 if this coefficient is equal to 1. 



The relation (1) should be satisfied by 

 all the values of y comprised in the field of 

 integration; it is therefore equivalent to a 

 continuous infinity of linear equations, 



Fredholm has treated the case of the 

 equations of the second kind; the solution 

 then may be put under the form of the 

 quotient of two expressions analogous to 

 determinants and which are integral func- 

 tions of A. For certain values of A, the 

 denominator vanishes. We then can find 



functions <i>{x) (called proper functions) 

 which satisfy the equation (1) when we 

 replace f{x) in it by 0. 



The result supposes that the kernel 

 K{x, y) is limited; if it is not, we are led 

 to consider reiterated kernels ; if we repeat 

 n times the linear substitution 8, we obtain 

 a substitution of the same form with a dif- 

 ferent kernel Kn{x, y) ; if one of these re- 

 iterated kernels be limited this suffices for 

 the method to remain applicable by means 

 of a very simple artifice. Now this hap- 

 pens in a great number of cases, as Fred- 

 holm has shown. The generalization for 

 the case where the unknown function de- 

 pends upon several variables and for that 

 where there are several unknown functions 

 is made without difficulty. 



Fredholm then applied his method to the 

 solution of Dirichlet's problem and to that 

 of a problem in elasticity, thus showing 

 how we may attack all questions of mathe- 

 matical physics. 



Such is the part of the first inventor; 

 what now is Hilbert's? Consider first a 

 finite number of linear equations; if the 

 determinant of these equations is sym- 

 metric, their first members may be re- 

 garded as the derivatives of a quadratic 

 form, and hence results for equations of 

 this form a series of propositions very 

 worthy of interest and well known to 

 geometers. The corresponding case for 

 integral equations is that where the kernel 

 is symmetric, that is to say, where 

 K{x, y)=K(y, x). 



This Hilbert takes hold of. The proper- 

 ties of quadratic forms of a finite number 

 of variables may be generalized so as to 

 apply to integral equations of this sym- 

 metric form. The generalization is made 

 by a simple passing to the limit; but this 

 passing presented difficulties which Hilbert 

 overcame by a method admirable in its 



