706 BELL SYSTEM TECHNICAL JOURNAL 



Now expniui the explicit solution as a power series in /: it is 



. _}_\at (aty , (at) 

 •'4 — „ -, , , 



i? ( 1! 2! ^ 3! J 



Comparing the two expansions we see at once that the operational 

 expansion is converted into the explicit solution by assigning to the 

 symbol l/p" the value t"/ti\. It was from this kind of inductive 

 inference that Heaviside arrived at his power series solution. 



Now there are several important features in the foregoing which 

 require comment. In the first place the operational equation is 

 converted into the explicit solution only by a particular kind of ex- 

 l)ansion, namely an expansion in in\erse powers of the operator p. 

 For example, if in the operational ecjuation 



R 1+a/p 

 we replace 1 p b\' t/ll we get 



1 at 

 R 1+0 



which is incorrect. I'^urthermorc, if we expand in ascending instead 

 of descending powers of p, naniel\- 



A=-^^\l-{p/a) + {p:ar— [ 



no ( orrelation with the explicit solution is possible and no significance 

 can be attached to the expansion. We thus infer the general princiiile, 

 and we shall find this inference to be correct, that the operational 

 equation is convertible into the explicit solution only by the proper 

 choice of expansion of the impedance function, or rather its reciprocal. 



In the second place we notice that in writing down the operational 

 equation and then converting it into the explicit solution no con- 

 sideration has been given to the question of boundary conditions. 

 This is one of the great advantages of the operational method: the 

 boundary conditions, provided lliey are those of equilibrium, are auto- 

 matically taken care of. This will be illustrated in the next example: 



l.el a "unit e.m.f." be impressed on a circuit consisting of resistance 

 K, inductance L, and capacity C: rc(]uiretl the resultant charge on the 

 condenser. 



The (lifferiMUial etiuation for the charge {) is 



{4+4t+'^c)Q-- 



1, 1^0. 



