414 „ . REPORTS ON THE STATE OF SCIENCE!. 



and an attempt is made to find a criterion for determining Whether this 

 condition is sufficient or not. 



The theory has been considerably extended by E. Schmidt. Ha 

 shows that the necessary and sufficient conditions for the solubility of 

 the equation consists in the convergence of a certain quadratic form. 



CO 



2 2S y M C E 



„=1 ] 1 = 1 



He constructs a set of orthogonal forms in the following way : 

 Let AJx) = a„ l . ■-!.•...«■ 



> < I = 1. 2 ... co 



Z{x) == Z x 



where a, u denotes the conjugate complex quantity to a ttX . 

 Put C 1 («) = A 1 («) 



C,(x) = A,(x) - 0,(a;) M22i£) 



C (x) = A M — Sf! M A »(-)^(-) 



then these forms C p ( x ) are orthogonal, and if 



D„(x) = C 1 (.r) + C 2 (.r) + . . . + C B (z) 

 the quantities y w are defined by the equation 



A.(a>) = 2 7l> D». 



If M(.r) is an arbitrary linear form, the form 



co 



P(a>) = M(«) - ia(*)M(.)C,(0 



is called the form (or function) perpendicular to the forms A r (.r) or C,(x). 

 Since the conditions of orthogonality are 



o.(W)~? v ^ 



" V ' M ' — 1 V — fl 



we have PQP^O = 



and this shows that P(.r) is orthogonal to all the C M 's. Schmidt gives 

 several different methods of solving the set of linear equations, but for 

 these we must refer to the original memoir. 



The importance of systems of linear equations with co' unknowns 

 depends upon the fact that an integral equation of the form 



h 



f(x) = 0(<c) — \L(x,t).i>{t,)Jt 



a 



or of the form 



f{x) = L{x,t)f{t)dt 



