CIRCUIT TIII.OKY .IXI> OI'P.KATIONAL CALCULUS 7S5 



powiTS of / is C()n\'erKciit while the series in ascending rx'^'i^s of t is 

 (liver^;ent : the converse is the case in the problems discussed previously. 

 A second specific problem may be stated as follows: 

 Let a "unit e.m.f." be impressed on an infinitely lon^ non-inductive 

 cable of distributetl resistance /?and capacity C per unit lenslh through 

 a terminal resistance /?..: rc(|uired the voltage Ton the cal)le terminals. 

 The formulation of the ojierational e(|uation of this problem is ver\' 

 simple. It will be recalled that the operatitjnal formula for the current 

 entering the cable with terminal voltage V is Vy/Cpj R. But the 

 current is clearly also equal to (1— V)/Ro: equating these expressions 

 we get 



^~^ =V\'pC7R 



Ro 



\\ tu'in 



V = —^ — (133) 



where 1/ v 'K = Roy/ C/ R. This is the required operational formula. 



To derive the Heaviside divergent expansion, expand (133) by the 

 binomial theorem : thus 



V=\-y/pI\ +(/>/X)-(p/X)3-''+ . . . 

 = l-(l+/./X + (/>/X)=+ . . )\/pJ\ (134) 



4-(/>/X + (/)/X)'+(/>/X)'+ . . . ). 



Discard the second series in integral powers of p\ replace y/p by \/y/rt 

 and p" by d" /dt" in the first series, thus getting 



^'-^~[^^\-di^r-dc-'^ ■■)v^i 



(135) 



which is the asymptotic solution of the problem. 



To verify this solution we shall consider the more general opera- 

 tional equation 



1 

 ^^ p'Wp+i ("'nlegral) (137j 



a form of equation to which a number of fairly important problems 

 is reducible. (The parameter X of equation (133) can be eliminated 

 from explicit consideration by means of theorem VI.) 



