and Conductance Operators. 481 



It is sometimes more convenient to make use of the converse 

 method. Thus, let Y be the reciprocal of Z, so that C = YV. 

 If we make p vanish in Y, the result, say Y , is the conduct- 

 ance of the combination. Therefore Y is the conductance- 

 operator. 



The fundamental forms of Y and Z are 



Z = R + Lp, (1) 



Y=K + Sp (2) 



In the first case, it is a coil of resistance R and inductance L 

 that is in question, with the momentum LC and magnetic 

 energy ^LC 2 . In the second case, it is a condenser of con- 

 ductance K and permittance S, with the charge SY and 

 electric energy ^SV 2 ; or its equivalent, a perfectly non- 

 conducting condenser having a shunt of conductance K. 



In a number of electromagnetic problems (no electric 

 energy) the resistance-operator of a combination, even a 

 complex one, reduces to the simple form (1). The system 

 then behaves precisely like a simple coil, so far as externally 

 impressed force is concerned, and is indistinguishable from a 

 coil, provided we do not inquire into the internal details. I 

 have previously given some examples*. Substituting con- 

 densers for coils, permittances for inductances, we see that 

 corresponding reductions to the simple form (2) occur in 

 electrostatic combinations (no magnetic energy). 



But such cases are exceptional ; and, should a combination 

 store both electric and magnetic energy, it is not possible to 

 effect the above simplifications except in some very extreme 

 circumstances. There are, however, two classes of problems 

 which are important practically, in which we can produce 

 simplicity by a certain sacrifice of generality. In the first 

 class the state of the whole combination is a sinusoidal or 

 simple harmonic function of the time. In the second class 

 we ignore altogether the manner of variation of the current, 

 and consider only the integral effects in passing from one 

 steady state to another, which are due to the storage of 

 electric and magnetic energy. 



3. If the potential-difference at the terminals be made sinu- 

 soidal, the current will eventually become sinusoidal in every 

 part of the system, unless it be infinitely extended, when con- 

 sequences of a singular nature result. At present we are 

 concerned with a finite combination. Then, if n/2ir be the 

 frequency of the alternations, we have the well-known pro- 



* " On the Self-induction of Wires," Parts VI. and VII. (Phil. Mag. 

 [5], vols, xxiii. & xxiv.) 



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



