and on Cductance Operators. 487 



ment of the steady state exceeds what would have been done 

 had this state been instantly established (but then without any 

 electric or magnetic energy) by twice the excess of the electric 

 over the magnetic energy. That is, 



i 



<fe2e(r-r >fc=2(U-T), .... (28) 



where e stands for an element of impressed force, T the 

 current-density at time t, T the final value, and 2 the space- 

 integration to include all the impressed forces. (Black letters 

 for vectors.) The theorem (28) seems the most explicit and 

 general representation of what has been long recognized in a 

 general way, that permitting electric displacement increases 

 the activity of a battery, whilst permitting magnetization 

 decreases it. The one process is equivalent to allowing elastic 

 yielding, and the other to putting on a load (not to increasing 

 the resistance, as is sometimes supposed). 



Applying (28) to our present case of one impressed force V, 

 producing the final current C, w r e obtain 



§dtV(T-C)dt=2(U-T), .... (29) 



comparing which w r ith (27), we see that 



T-U=iZ 'C 2 =iL C 2 =-iS V 2 , . . . (30) 



confirming the generality of our results. 



8. It is scarcely necessary to remark that the properties of 

 Z and 7J previously discussed do not apply merely to com- 

 binations consisting of coils of fine wire and condensers ; the 

 currents may be free to flow in conducting masses or dielectric 

 masses. Solid cores, for example, may be inserted in coils 

 within the combination . The only effect is to make the 

 resultant resistance-operator at a given place more complex. 



But a further very remarkable property we do not recognize 

 by regarding only common combinations of coils and con- 

 densers. If w r e, in the complex medium above defined, select 

 any unclosed surface, or surface bounded by a closed line, and 

 make it a shell of impressed force (analogous to a simple mag- 

 netic shell), thereby producing a potential-difference Y between 

 its two faces, and C be the current through the shell in the 

 direction of the impressed force, there must be a definite 

 resistance-operator Z connecting them, depending upon the 

 distribution of conductivity, permittivity, and inductivity 

 through all space, and determinable by a sufficiently ex- 

 haustive analysis. The remarkable property is that the 

 resistance-operator is the same for any surfaces having the 

 same bounding-edge. For a closed shell of impressed force 



