and Conductance Operators. 497 



tiating it, we find 



Lo ~^(RKy 



O-K) w 



corroborating the previous result as to the vanishing of L 

 when the circuit is infinitely long by equality of RS and KL 

 and the positivity of L when KL > US. 



12. Now, in this singular case, we have, by (41) and (42), 



Z = Lv, L =0, (43) 



if v = (LS)~% the speed of transmission of disturbances along 

 the circuit. The resistance-operator has reduced to an absolute 

 constant, and the current and potential-difference are in the 

 same phase, altogether independent of the frequency of alter- 

 nations, or indeed of the manner of variation. The quantity 

 Lu, or L x 30 ohms, approximately, if the dielectric be air, is 

 strictly, and without any reservation, the impedance of the 

 circuit at A, but it is only exceptionally the resistance. 



Make V =/(£), at A, an arbitrary function of the time; 

 then, if V* and 0* are the potential-difference and the current 

 at distance x from A at time t, respectively, we shall have 



V«=/(*-*/»)e-^ | 



G x =VJLv V 



or all disturbances originating at A are transmitted un- 

 distorted along the circuit at the speed v, attenuating at 

 a rate indicated by the exponential function. (I have else- 

 where* fully developed the properties of this non-distortional 

 circuit, and only mention such as are necessary to understand 

 the peculiarities connected with the present subject-matter.) 

 The electric and magnetic energies are always equal, not only 

 on the whole, but in any part of the circuit ; this accounts for 

 the disappearance of L , and the bringing of V* and C* to the 

 same phase, as we should expect from § 4. But in the present 

 case Z , or Lv, or B/, for they are all equal, is only the resistance 

 when the steady state due to the steady V at A is arrived at 

 (asymptotically) , or the effective resistance at a given frequency 

 when Y is sinusoidal, and sufficient time has elapsed to have 

 allowed Y^, and C* to become sinusoidal to such a distance 

 from A that we can neglect the remainder of the circuit 

 into which greatly attenuated disturbances are still being 

 transmitted* 



13. Now, since the impedance is unaltered by joining on 

 at A any length of circuit of the same type, and is a constant, 



* " Electromagnetic Induction and its Propagation," Arts, xl. to l., 

 • Electrician/ 1887. 



Phil. Mag. S. 5. Yol. 24. No. 151. Dec. 1887. 2 L 



