718 BELL SYSTEM TECHNICAL JOURNAL 



problems, is one which permits us to infer the general character and 

 properties of the function and the effect of the circuit constants on 

 its form, without detailed solutions. A solution which possesses 

 these properties, even if its exact computation is not possible without 

 prohibitive labor, is far superior to a solution which, while com- 

 pletely computable, tells us nothing without detailed computation. 

 It is for this reason that some of the derived forms of solution, dis- 

 cussed later, are of such importance. In fact a solution which re- 

 quires detailed computation before it yields the information implied 

 in it is merely equivalent to an experimentally determined solution. 

 Unfortunately the advantages attaching to the expansion solution 

 of the specific problem just discussed, do not, in general, characterize 

 the expansion solution. The following disadvantages should be 

 noted. First, the location of the roots of the impedance function 

 H{p) is practically impossible in the case of arbitrary networks of 

 more than a few degrees of freedom. In the second place, when the 

 number of degrees of freedom is large it is not only impossible to 

 deduce the significance of the solution by inspection, but the com- 

 putation becomes extremely laborious. In such cases, the practical 

 value of the expansion solution depends, just as in the power series 

 solution, on the possiljility of recognizing and summing the expan- 

 sion. This will be clear in the case of transmission lines, where the 

 roots of H{p) are infinite in number and the direct computation of 

 the expansion solution (except in the case of the non-inductive cable) 

 is C|uiti' impnssil)k'. 



CHAPTER IV 



.SoMic Genkrai. Formulas and Theorems for the .Solution 

 OE Operational Equations 



W'e have seen lli.il tlic operational equation 



h = \/II{p) 

 is the symbolic or siiort-liand einiixalent (jf the integral ecjuation 



and from the latter we have deduced two very important forms of the 

 Heaviside solution. In recognizing the equi\alence of these two 

 equations we have a very great advantage and are able, in fact, to 

 base the Operational Calculus on deductive instead of inductive 



