ON 'fH£ THEORY OF INTEGRAL EQUATIONS. 413 



intervals between the bands. A certain limited form <r(\ ; x,x), called 

 the spectral form is associated with each point of a band, and the 

 equations (3), (4), and (2) are now replaced by 



S 



(L p (x)] 2 fdrT(n ] X,x) 



K(\ ; x,x) = ^ -i A. 



+ i~* 



CO /• 



(xx) = s [ii B {x)f + ^0" ; &*)■ 



p=i j 



The quantities \ p can also possess points of condensation in the 

 finite part of the axis of real quantities. 



The proofs of these general theorems are very difficult, some of the 

 results have, however, been established in a simpler way by Hilbert's 

 followers. 1 The literature on the subject is now quite extensive. 



25. Linear Equations in an Infinite Number of Variables. 



The theory of linear equations with an infinite number of unknowns 

 which was first considered by G. W. Hill, 2 Poincare, 3 and Helge von 

 Koch 4 in connection with the theory of infinite determinants has been 

 the subject of some recent investigations by Hilbert, and the results 

 have been simplified and extended by Toeplitz. The equations con- 

 sidered are of the form 



CO 



2rt„ m 4 - C„ . . . (l) 



m=l (n — I, 2 i . i tooo) 



and attention is paid only to the solutions for Which the series 



m = l 



is convergent, 



Hilbert and Toeplitz discuss the case when the quantities a,,, n are the 

 coefficients of a limited bilinear form and the series 



fco / / ., 



2/ U>nml 



m = i 



converges for" all values of n. Then the convergence of the series 



fcd / / 



a/OL/a 



is assumed as a necessary condition for the solubility of the equations, 



1 Reference may be made to the Gb'ttingen dissertations by Hellinger (1907) and 

 Weyl (1908); and papers by Hellinger and Toeplitz {Gottingen Nachrichten, 1906). 

 Hellinger {Crelh, Dd. 13G, 1909), Toeplitz {Rend. Palermo), Plancherel {Math. Ann., 

 Ud. 67, p. 5tl). RiessS {Gott. JVachr.. 1910), Hilbert {Gott. Nachr., 1910). 



1 Acta Math., 1886, t. 8, pp. 1-36. The memoir was first published in 1877. 



* Bud. de la, Sue. Math, de France, 1886, t. 14, pp. 77-90. 



' Acta Math., 1891, t. 15, pp. 53-63 ; ibid., 1892-93, t. xvi., pp. 217-295. 

 1910. B E 



