ON THE THEORY OF INTEGRAL EQUATIONS. 399 



Hadamard ' has proved that any linear operation A(a) can be expressed 

 in the form 



b 



A(«) = lim \K„(x)a(x)dx 



n ->co J 



where K n (x) is a suitably chosen function. This result is important 

 because it indicates the general form to be expected for the inversion 

 formula of a linear integral equation. 



Frechet 2 has given other proofs of Hadamard's theorem, and has 

 shown that in particular the functions K„(x) may be taken to be 

 polynomials. 



The transformation of operations is also of some importance in the 

 theory of integral and differential equations. Let X be an operation 

 which can be inverted in a unique manner by means of an operation 

 X" 1 , then an operation A is said to be transformed in B by means of X 

 when we have identically 3 



B = XAX- 1 . 



If A, is transformed into Bj and A 2 into B 2 , then A 2 A 2 is trans- 

 formed into B.B^ If the operations A form a group, the operations B 

 also form a group. If two of the A's are permutable the corresponding 

 B's are also permutable. If A is transformed into B the adjoint operation 

 B is transformed into A. These theorems find a good illustration in 

 the connection between the transformations of Euler and Laplace in 

 the theory of linear differential equations. 



19. Connection with the Calculus of Variations. 



A connection between integral equations and the calculus of varia- 

 tions was indicated by Volterra 4 in 1884. He showed that the integral 

 equation 



f(x)=y-(x,t)f(t)dt o<x±a 



o 



may be obtained by making the first variation of the quantity 



a a a 



P = - 2 -j | K(x,fy(x)t(Qdxdt - f(x)f(x)dx 



equal to zero. 



Hilbert obtained another connection by showing that the maximum 

 value of the integral 



h h 



| U*,ty(x)t(t)dxdt 



a a 



1 Hadamard, Conipt"s rendvs, Feb. 9, 1903. 



• Trans. Amer. Math. Son., 1905, vol. vi., No. 2, p. 138. 



3 Pinchcrle and Amaldi, Ch. 1 3. The theory was applied to continuous groups of 

 transformations by C. Jordan. Annali di Matematica, 1SG8, ser. ii. t 2 



4 II nuoro Cimento, 1884, series 3 A , vol. xvi., p. 49. 



d d 2 



