ON THE THEORY OP INTEGRAL EQUATIONS. 411 



tions connected with the problem are very long, and so we must refer to 

 the original memoirs. 



A singular integral equation of the second kind of the type 



CO CO 



f(x) = B(x) + 1 j" <j,(t)dt [ 0(£) cos *($ - x)di 



— co — CO 



was obtained and solved by 0. Tedone l in the problem of the elastia 

 equilibrium of an indefinite circular cylinder. 

 The solution is given by the formula 



CO CO 



24. Bilinear and Quadratic Forms in an Infinite Number of Variables. 



The theory of quadratic forms in an infinite dumber of variables is 

 propounded in a memoir by Hilbert, 2 and is applied to integral equations 

 in a subsequent memoir. These memoirs contain many theorems of a 

 fundamental nature which are of a very wide application. The theoiyis 

 interesting, as it is full of striking analogies, the occurrence of point and 

 band spectra being particularly noteworthy. Let 



co 



k m = «.-«, ^x,r = (xx) < 1, ()Jlj) < 1. 



n = 1 



then the expression 



CO CO 



l<[X, X) = ^j ZjL'pqX^Xq 



t> = l q=l 



is called a quadratic form in an infinite number of variables, and the 

 expression 



CO CO 



k(x, V ) = 2 2< vq x r tf 



a bilinear form. When k- pq = -j _ the bilinear form is written (xy). 



The bilinear form is said to be limited when it remains below a fixed 

 limit for all values of x and y which satisfy the conditions (xx) < 1 (yy) < 1. 

 If k(x,y) is limited, so also is k(x,x) and vice versa. If 2c 2 ;) , 2 converges, 

 k(x,y) is called a limited continuous (vollstetig) form. pq 



n n 



The expression K n (x,y) = 2 23 ic m x p y q 



p=l 3=X 



is called the n"' section of k(x,y). 



If k(x,y) is a limited form, ic„(x,y) converges for fixed values of x,y to 

 ic(x,y) as n -» co, provided (xx) < 1 (yy) < 1. The convergence to the limit 

 is uniform provided the series 2#/ 2 , Eyf converge uniformly. 



1 Bend. Lincei, 1904, p. 232. 2 Gottingcn Nadir iclit en, 1906. 



