ELECTRIC CIRCUIT THEORY 69 



The preceding solution depends for its outstanding directness 

 and simplicity on the recognition of the infinite integral identity (m), 

 into which the integral equation of the problem can be transformed. 

 When such identities are known their value in connection with the 

 solution of operational equations requires no emphasis. On the 

 other hand, we cannot always expect to find such an identity in the 

 case of every operational equation; and, particularly in the case of 

 such an important case as the transmission equation it would be 

 unfortunate to have no alternative mode of solution. Fortunately 

 a quite direct series expansion solution is obtainable from the oper- 

 ational equation, and this will now be derived. As a matter of con- 

 venience we shall restrict the derivation to the voltage formula 



7=e-7V(^+^)^^. (203) 



As a further matter of mere convenience we shall assume that G = 0, 

 so that (T = p and (203) becomes 



y^g-T^|p^+2pp (203-a) 



where t = x/v. 



The method holds equally well for the current equation (204) and 

 for the general case a^^p. 



Write (203-a) as 



and expand the exponential factor {l-\-2p/py^^ by the binomial 

 theorem; thus 



(l-f2p//,)'/2 = l+|+«,(^)V«3(^)'+ . . . 



so that 



Now the operational equation 



/ aiTfp- asTp^ atTp* \ 



can be expanded in inverse powers of p; thus 



p p 



the power series solution of which is 



«' = i + ^ + ir2 + X3 + 





