412 REPORTS ON THE STATE OP SCIENCE. 



The fold (Faltung) of two linear forma 



CO CO 



L(x) = n&i U(.v) = ZniiXt 



= 1 =1 



is defined by the equation 



L(.)M(0^2^ 



i 



A linear form 2^ is said to be limited when 21? converges. For linear 1 

 forms we have the inequality 



(2U) 2 <(2^)(2^). 



i i I 



When k(x,x) is a limited continuous quadratic form the solution of 

 the equation 



L i> (y) = \ p Z p (-y(.,y) . . . . (1) 



is possible by a limited linear form L,,(>/) for certain quite definite real 

 values of \. These values of A„, which in the present case can only 

 condense at infinity, 1 are called the characteristic values (Eigenwcrlc), and 

 Jj p {y) the characteristic forms. The latter can be chosen so that 



h(')h(') = if q±p, 



L,(-)L.(-) = 1 



A set of linear forms which satisfy these equations are said to form 

 an orthogonal system. 



We have the relations 2 



.(x,,) = 1 M 



. . . . (2) 



P A P 



CO 



(xx) = Z[L p (x)? .... (3) 



The aggregate of the characteristic values A,, is called the spectrum 

 of the quadratic form. For each value of y outside the spectrum there 

 exists a uniquely defined resolvent 



K(X; x,x)=:Z , x ... (4) 



which satisfies the equation 



K(A; x^-Hx, • )K(\; . ,y) = {xy) 



If K(x,y) is a limited but no longer a continuous form, the equation 

 (1) is generally satisfied by an aggregate of real values of X, n which are 

 arranged in continuous bands, and there may be isolated points in the 



1 The value A p = <» may enter either simply or multiply among the characteristic 

 values. 



8 The resolution of a quadratic form into a sum of squares has been further 

 developed by Toeplitz and von Koch {Math. Ann., 1910, Bd. 69, p. 26fi). The 

 latter gives a method depending on the use of infinite determinants by which the 

 reduction may be actually effected. 



