ELECTRIC CIRCUIT I IIEORY 361 



A power series solution, when it exists, can be gotten by repeated 

 differentiation; tlius 



4>{o)=I{o), 



<^'(x) =^f\x)+<i>{o)K{x)+ I '^4>'{x-y)K{y)dy, 



^'{o)=r{o)+<i.{o)K{o) 



In this way all the derivatives at .r = are calculable; let them be 

 denoted by 0o, 4>i, 02 ■ . • Then 



This form of solution, also, is of limited practical usefulness, except 

 for small values of x. 



A number of mathematicians, including Wittaker and Bateman, 

 have studied the question of numerical solution and suggested other 

 processes. After quite extensive study of the question, however, the 

 writer is of the opinion that point-by-point numerical integration 

 like that discussed in the text is, in general, the most practical, rapid 

 and accurate method of numerical solution. This judgment is con- 

 firmed by G. Prasad who, in a paper on the Numerical Solution of 

 Integral Equations delivered before the International Mathematical 

 Congress (Toronto, 1924), discusses the whole question and arrives 

 at the same conclusion. 



In the text, numerical integration is carried out in accordance 

 with Simpson's Rule. It is possible, of course, to employ more com- 

 plicated and refined formulas for approximate quadrature. It is the 

 writer's opinion that this is hardly justified in practical problems and 

 that the required accuracy is more simply obtained by employing 

 smaller intervals. 



Appendix to Chapter IX. Note on Conventions as to Signs in Netivorks 



In the network shown on page 196 the arrows indicate the direc- 

 tions chosen as positive in the network itself, quite regardless of the 

 presence of any e.m.fs. and currents. 



The sign attributed to a current, an e.m.f., or a voltage is positive 

 if the current, e.m.f., or voltage is in the positive direction; otherwise 

 the sign is negative. 



Stated more fully: 



A current at a specific point (at a specific instant of time) is posi- 



