360 BELL SYSTEM TECHNICAL JOURNAL 



ing the distant termination. Evidently, from physical considerations, 

 the series of component vibrations making up the complete solution 

 must therefore so combine as to annihilate the efifect of the distant 

 termination for a finite time. The solution is, therefore, mathe- 

 matically correct but physically artificial. 



Note on Integral Equations. 



An integral equation is defined as an equation in which the unknown 

 function occurs under a sign of integration; the process of determining 

 the unknown function is called solving the equation. 



Integral equations are of great importance in mathematical physics 

 and in recent years very considerable work has been done on them 

 from the standpoint of pure analysis. 



The types of integral equations with which we are concerned in 

 the present work are Laplace s Equation 



F{p)= r e-P'mdt 

 Jo 



and Poisson's Equation 



ct>(x) =f{x) + / ''4>(y)K(x - y)dy. 

 ./o 



But little work has been done on Laplace's Equation from the 

 standpoint of pure analysis; its most extensive and useful applications 

 appear to be in connection with the Operational Calculus. Practical 

 methods of solution are extensively discussed in the text. 



We shall now briefly discuss the solution of Poisson's Equation. 



The formal series solution, which is absolutely convergent, is ob- 

 tained by successive substitution. Thus suppose we write 



(/)(.r)=(^o(.x)+</>i(-^-)+02(.'v)+ 



and define the terms of the series in accordance with the scheme 



4>i{x)^ r\(y)K{x-y)dy, 

 Jo 



4>2(x)= I <t)l(^•)K{x-y)dy,etc., 

 Jo 



the resulting series satisfies the integral e(iuation and is absolutely 

 convergent. It is, however, practically useless for cominUalion or 

 interpretation. 



