and Conductance Operators. 499 



after starting it ; then j^Lv for a second period 21/ v; then JLv 

 for a third period, and so on. 



It will have been observed that I have, in the last four para- 

 graphs, used the term impedance in a wider sense than in § 3, 

 where it is the ratio of the amplitude of the impressed force 

 to the amplitude of the flux produced at the place of impressed 

 force when sufficient time has elapsed to allow the sinusoidal 

 state to be reached, when that is possible. The justification 

 for the extension of meaning is that, since in the non-distor- 

 tional circuit of infinite length, or of finite length with a 

 terminal resistance to take the place of the infinite extension, 

 we have nothing to do with the frequency of alternations, or 

 with waiting to allow a special state to be established, it is 

 quite superfluous to adhere to the definition of the last sen- 

 tence ; and we may enlarge it by saying that the impedance 

 of a combination is simply the ratio of the force to the flux, 

 when it happens to be a constant, which is very exceptional 

 indeed. I may add that R, L, K, and S need not be constants, 

 as in the above, to produce the propagation of waves without 

 tailing. All that is required is R/L = K/S, and ~Lv = constant; 

 so that R and L may be functions of x. The speed of the 

 current, and the rate of attenuation, now vary from one part 

 of the circuit to another. 



16. In conclusion, consider the application of the resistance- 

 operator to normal solutions. If we leave a combination to 

 itself without impressed force, it will subside to equilibrium 

 (when there is resistance) in a manner determined by the 

 normal distributions of electric and magnetic force, or of 

 charges of condensers and currents in coils ; a normal system 

 being, in the most extended sense, a system that, in subsiding, 

 remains similar to itself, the subsidence being represented by 

 the time-factor e pt , where p is a root of the equation Z = 0. It 

 is true that each part of the combination will usually have a 

 distinct resistance-operator ; but the resistance-operators of 

 all parts involve, and are contained in, the same characteristic 

 function, which is merely the Z of any part cleared of frac- 

 tions. It is sometimes useful to remember that we should 

 clear of fractions, for the omission to do so may lead to the 

 neglect of a whole series of roots ; but such cases are excep- 

 tional and may be foreseen ; whilst the employment of a 

 resistance-operator rather than the characteristic function is 

 of far greater general utility, both for ease of manipulation 

 and for physical interpretation. 



Given a combination containing energy and left to itself, it 

 is upon the distribution of the energy that the manner of sub- 

 sidence depends, or upon the distribution of the electric and 



2 L 2 



