708 BELL SYSTEM TECHNICAL JOURNAL 



where kx and ki are constants which must be chosen to satisfy the 

 boundary conditions and Xi, Xo are the roots of the equation 



L\- + R\ + ]. C = o. 



Now since we have two arbitrary constants we satisfy the equilibrium 

 condition by making Q and dQ/dl zero at i = o, whence 



C+kl + k2=0, 

 Xl^l+X2^2=0, 



and 



We ha\'e also 



)^i = X2C/(Xi-X,), 

 /to = XiC/(Xo-X,). 



x.--fW(f)'->' 



Writing down the power series expansion of 

 then 



+ (^iXr + ^>X.r)^"j+ 



Introducing the values of ki, ki, Xi, Xj given above and comparing 

 with the power series derived from the operational solution we see 

 that tiiey are identical term by term. 



This example illustrates two facts. First the power series expansions 

 may be complicated, laborious to derive and of such form that they 

 cannot be recognized and summed by inspection. In fact in arbitrary 

 networks of a large number of meshes or degrees of freedom the 

 evaluation of the coefficients of the power series expansion is extremely 

 laborious. 



On the other hand, in such cases, the solution by the classical 

 method presents difficulties far more formidable — in fact insuperable 

 difficulties from a practical standpoint. First there is the location 

 of the roots of the function //(X), which in arbitrary networks is a 

 pnictical impossibility without a prohibitive amount of labor. Sec- 

 ondly there is the determination of the integration constants to satisfy 

 the imposed boundary conditions: a process, which, while theoretically 



