ON THE THEORY OP INTEGRAL EQUATIONS. 397 



closed. These definitions have been extended by E. Sohmidfc ' to func 

 tions of an infinite set of variables. 



A field of functions is said to be linear if, when a, ji are any twc 

 functions belonging to the field, the function 



X« + fift 



also belongs to the field, \ and n being arbitrary constants. 



It often happens that by performing an operation A upon a function 

 a of a field M another function (i of the field is produced. When this 

 is the case the functions for which the operation is periodic are of 

 special interest, in particular if the operation is of period one so that 



o = XA(«) 



where X is a constant the function a is called an invariant of the 

 operation A (Pincherle). Thus in the case of the operation defined by 

 the definite integral 



b 



Mf) = Us,t)f(t)dt 



a 



the invariants <j> satisfy integral equations of the type 



b 

 <j(s) =! X Us,t)l>{t)dt. 

 a 



If A and B are two operations the equation 



AB/» = BA^> 



is generally a functional equation which determines the functions </> with 

 respect to which the operations A, B are commutative, but it may 

 happen that it is satisfied identically. In the case of the operations 



b 



A(/) = UsJWM B(/) = [h(s,t]f(t)dt 



this is the case if 2 



b 



f'M*W)<fc = f*M*^. 



This equation is satisfied by series of the type 



h(x,t) = Za n t n {x) x ,Xt) 

 where </>„(.r), x„(t) satisfy the homogeneous integral equations 



M'«( s ) = U{s,x)^ n (x)dx 



b 



1 Rend. Palermo, 1907. 



J Some properties of perrnutable functions arc coiisitlerel by Volterra, Bend 

 Lined, April 17, 1910. 



1910. u d 



