804 



SGIENGE 



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



this decomposition when the number of 

 variables becomes infinite? The poles of 

 the function rational in A may in this case 

 or otherwise tend toward certain limit 

 points infinite in number but discrete. 



The aggregate of these points consti- 

 tutes what our author calls the discontinu- 

 ous spectrum of his form. They may also 

 admit as limit points all the points of one 

 or several sects of the real axis. The ag- 

 gregate of these sects constitutes the con- 

 tinuous spectrum of the form. 



The simple fractions corresponding to 

 the discontinuous spectrum will make in 

 their totality a convergent series; those 

 corresponding to the continuous spectrum 

 will change at the limit into an integral of 

 the form 



/: 



Orf/i 



where the variable of integration jx is 

 varied all along the sects of the continuous 

 spectrum, and where o- is a suitable func- 

 tion of /i. The rational function K{\,x,y), 

 therefore, has then as limit not a mero- 

 morphic function, but a uniform function 

 with erasures. The decomposition into 

 simple elements thus transformed remains 

 valid. If the given form is limited, that is 

 to say, if it can not pass a certain value 

 when the sum of the squares of the vari- 

 ables is less than 1, we can deduce thence a 

 way of simplifying this form by an orthog- 

 onal transformation, analogous to the 

 simplification of the equation of an ellip- 

 soid by referring this surface to its axes. 



Among the quadratic forms we shall dis- 

 tinguish those which are properly continu- 

 ous (vollstetig) , that is to say, those whose 

 increment tends toward zero when the in- 

 crements of the variables tend simultane- 

 ously toward zero in any way. Such a 

 form does not have a continuous spectrum 

 and hence result noteworthy simplifica- 

 tions in the formiilas. 



Other theorems on the systems of simul- 

 taneous quadratic forms, on bilinear forms, 

 on Hermite's form, extend likewise to the 

 case of an infinite number of variables. 



There was in this theory the germ 

 of an extension of Fredholm's method to 

 kernels to which the analysis of the Swed- 

 ish geometer was not applicable, and schol- 

 ars of Hilbert should bring out this fact. 

 However that may be, Hilbert first applied 

 himself to extending his way of looking at 

 integral equations to the cases where the 

 kernel is unsymmetrie. For this purpose 

 he introduces any system of orthogonal 

 functions, conformably to which it is pos- 

 sible to develop an arbitrary function by 

 formulas analogous to that of Fourier. In 

 place of an unknown function, he takes as 

 unknowns the coefficients of the develop- 

 ment of this function; an integral equa- 

 tion can thus be replaced by a system of a 

 discrete infinity of linear equations be- 

 tween a discrete infinity of variables. 



The theory of integral equations is thus 

 attached, on the one hand, to the ideas of 

 von Koch on infinite determinants, and, on 

 the other hand, to the researches of Hilbert 

 we have just analyzed and where the essen- 

 tial role is played by functions dependent 

 upon a discrete infinity of variables. 



To each kernel will correspond thus a 

 bilinear form dependent upon an infinity 

 of variables. If the kernel is symmetric, 

 this bilinear form is symmetric and may be 

 regarded as derived from a quadratic form. 

 If the kernel satisfies the conditions stated 

 by Fredholm, we see that this quadratic 

 form is properly continuous and conse- 

 quently does not have a continuous spec- 

 trum. This is a way of reaching Fred- 

 holm's results, and however indirect it may 

 be, it opens entirely new views of the pro- 

 found reasons for these results and hence 

 on the possibility of new generalizations. 



Integral equations lend themselves to the 



