Mat 26, 1911] 



SCIENCE 



803 



simplicity, certainty and generality. The 

 developments reached are uniformly con- 

 vergent, but this "uniformity presents itself 

 under a new form which deserves to at- 

 tract attention. In the developments 

 appears an arbitrary function u{x) (or 

 several) and the remainder of the series 

 when n terms have been taken is less than 

 a limit depending only upon n and inde- 

 pendent of the arbitrary function, provided 

 this function is subject to the inequality 



ru-(x')dx < 1, 



the integral being taken between suitable 

 limits. This is an entirely new considera- 

 tion which may be utilized in very different 

 problems. 



Thus Hilbert obtains in a new way cer- 

 tain of Fredholm's theorems; but I shall 

 stress above all the results which are most 

 original. 



In the first place, the denominator of 

 Fredholm's expressions is a function of A. 

 admitting only real zeroes, and this is a 

 generalization of the elementary theorem 

 relative to "the equation in /S." After- 

 ward comes a formula where enter under 

 the sign f two arbitrary functions x{s) 

 and y{s) which we should consider as the 

 generalization of the elementary formulas 

 which permit the breaking up of a quad- 

 ratic form into a sum of squares. 



But I hasten to reach the question of the 

 development of an arbitrary function pro- 

 ceeding according to proper functions. Is 

 this development, the analogue of Fourier's 

 series or of so many other series playing a 

 principal role in mathematical physics, 

 possible in the general case 1 The sufficient 

 condition that a function be capable of 

 such development is that it can be put in 

 the form 8g{x), g{x) being continuous. 

 This is the final form of the resultant as 

 Hilbert gives it in his fifth communication. 

 In the first he was forced to impose certain 



restrictions ; here we must mention the 

 name of Schmidt, who in the interval had 

 produced a work which helped Hilbert to 

 free himself from these restrictions. The 

 only condition imposed upon our function 

 is capability of being put in the form 

 8g{x), and at first blush this would seem 

 siifficiently complex, but in a large number 

 of cases and, for example, if the kernel is a 

 Green's function, it only requires that the 

 function possess a certain number of de- 

 rivatives. 



Hilbert was afterward led to develop 

 his views in the following manner: he this 

 time considers a quadratic form with an 

 infinite number of variables and he studies 

 its orthogonal transformations; this is as 

 if he wished to study the different forms of 

 the equation of a surface of the second de- 

 gree in space of an infinite number of 

 dimensions when referred to different 

 systems of rectangular axes. To this 

 effect he makes what he calls the resolvent 

 form of the given form. Let K{x) be 

 the given form, E{\, x, y) the resolvent 

 form sought ; it will be defined by the iden- 

 tity 



K{\ x,y)—ir. 



dK[x) dK{\ X, y) 



dZr dXr 



■ ^XrVr. 



When the form K{x) depends only upon 

 a finite number of variables, the resolvent 

 form presents itself as the quotient of two 

 determinants which are integral polynom- 

 ials in A. 



Our author applies to this quotient the 

 procedures of passing to the limit which 

 are familiar to him; the limit of the 

 quotient exists even when those of the 

 numerator and of the denominator do not 

 exist. 



In the case of a finite number of vari- 

 ables, K{\, X, y) is a rational function of 

 A and this rational function can be broken 

 up into simple fractions. What becomes of 



