and Conductance Operators. 495 



The following illustration of the properties of Z and Z ' is 

 a complex one, but I choose it because of its comprehensive 

 character, and because it leads to some singular extreme 

 cases, interesting both mathematically and in the physical 

 interpretation of the apparent anomalies. Let the combination 

 be a telegraph -circuit, say a pair of parallel copper wires, of 

 length I ; resistance R, permittance S, inductance L, and 

 leakage-conductance K, all per unit length, and here to be 

 considered strictly constants, or independent of p. Let the 

 two wires be joined through an arrangement whose resistance- 

 operator is Z x at the distant end B ; then the resistance- 

 operator at the beginning A of the circuit is given by* 



z= (R+Lp)l{ (tanm/)/mZJ- +Zj ^ 



l + tK + S^ZZx^tanmO/M)^ [ } 



if -m 2 = (R + Lp)(K+Sp) (33) 



Take Z x =0 for the present, or short-circuit at B. This 

 makes 



Z=(R + Lp)Z(tanmZ)/mf, . . . (34) 

 and the steady resistance at A is therefore 



Z =E.Z(tanw2 Z)/??2 Z, (35) 



if -m 2 = EK. Also, differentiating (34) to p, and then 

 making p = 0, we find 



Z/=L =iZ^(L-^)+i«sec»w(L + ^) . (36) 



represents the impulsive inductance. 



If we put S = in (36) we make the arrangement electro- 

 magnetic, and then L is positive. If we put L = 0, we make 

 it electrostatic, and L is negative, or S , the impulsive per- 

 mittance, is positive. It is to be noticed that there is no con- 

 fusion when both energies are present ; that is, there are no 

 terms in Z ' containing products of real permittances and 

 inductances, which is clearly a general property of resistance- 

 operators, otherwise the two energies would not be inde- 

 pendent. 



We may make L vanish by special relations. Thus, if there 

 be no leakage, or K = 0, (36) is 



L =LZ-iRZ.BS/ 2 ; (37) 



so that the electromagnetic must be one third of the electro- 

 static time-constant to make the " extra-current " and the 

 static charge balance. (The length of the circuit required for 



* " On the Self-induction of Wires," Part IV., Phil. Mag. Nov. 1886. 



