712 BELL SYSTEM TECHNICAL JOURNAL 



In other cases the form of the generalized impedance function H{p) 

 will indicate by inspection the appropriate procedure. A general 

 process, applicable in all cases where a power series exists, is as follows. 

 Write 



]///(/>) = 1/7/(1) =G(.v). (36) 



Now expaiul (7(.v) as a Ta>l()r's series: thus formalK' 



GW=C(o) + G(')(o)-j^+G(2)(,,:):|+ . . . 



where 



GW(o) = [^G(.r)]^_^. (37) 



G(«)(o) 

 Denote j — by a„, replace x" by l//>", and wo lia\x' 



G(.v) = l///(/>)=ao+a,/p+fl2//>2+ . . . 



This process of "algebrizing" is formally straightforward and always 

 possible. As implied above, however, in many problems much shorter 

 modes of expansion suggest themselves from the form of the function 



mp). 



We note here, in passing, that the necessary and sufficient condi- 

 tions for the existence of a power series solution is the possibility of 

 the formal expansion of G{x) as a power series in x. 



At this point a brief critical estimate of the scope and value of the 

 power series solution may be in order. As stated above, in a certain 

 important class of problems relating to transmission lines, a power 

 series does not exist, though a closely related series in fractional 

 powers of t may often be derived. Consequently the power series 

 solution is of restricted applicability. Where, however, a power 

 series docs exist, in directness and simplicity of derivation it is superior 

 to any other form of solution. Its chief defect, and a very serious 

 defect indeed, is that except where the power series can be recog- 

 nized and summed, it is usually practically useless for computation 

 and interpretation except for relatively small values of the time /. 

 This disadvantage is inherent and attaches to all power series solu- 

 tions. For this reason I think Heaviside overestimated the value 

 of power series as practical or working solutions, and that some of 

 his strictures against orthodox mathematicians and their solutions 

 may be justly urged against the power series solution. He was C|uile 

 right in insisting that a solution must be capable of cither interpre- 

 tation or computation and quite right in ridiculing those formal 



