480 Mr. 0. Heaviside on Resistance 



electromagnetic combination be imagined to be cut anywhere, 

 producing two electrodes or terminals. Let the current enter- 

 ing at one and leaving at the other terminal be C, and let the 

 potential-difference be V, this being the fall of potential from 

 where the current enters to where it leaves. Then, if V = ZC 

 be the differential equation (ordinary, linear) connecting V 

 and C, the resistance-operator is Z. 



All that is required to constitute a self-contained system is 

 the absence of impressed force within it, so that no energy 

 can enter or leave it (except in the latter case by the irrever- 

 sible dissipation concerned in Joule's law) until we introduce 

 an impressed force ; for instance, one producing the above 

 potential-difference V at a certain place, when the product 

 VC expresses the energy-current, or flux of energy into the 

 system per second. 



The resistance-operator Z is a function of the electrical 

 constants of the combination and of d/dt, the operator of time- 

 differentiation, which will in the following be denoted hyp 

 simply. As I have made extensive use of resistance-operators 

 and connected quantities in previous papers *, it will be suf- 

 ficient here, as regards their origin and manipulation, to say 

 that resistance-operators combine in the same way as if they 

 represented mere resistances. It is this fact that makes them 

 of so much importance, especially to practical men, by whom 

 they will be much employed in the future. I do not refer to 

 practical men in the very limited sense of anti- or extra- 

 theoretical, but to theoretical men who desire to make theory 

 practically workable by the simplification and systematization 

 of methods which the employment of resistance-operators and 

 their derivatives allows, and the substitution of simple for 

 more complex ideas. In this paper I propose to give a con- 

 nected account of most of their important properties, including 

 some new ones, especially in connexion with energy, and some 

 illustrations of extreme cases, which are found, on examina- 

 tion, to "prove the rule." 



2. If we put ^> = in the resistance-operator of any system 

 as above defined, we obtain the steady resistance, which we 

 may write Z . If all the operations concerned in Z involve 

 only differentiations, it is clear that when C is given com- 

 pletely, V is known completely. But if inverse operations 

 (integrations) have to be performed, we cannot find Y from 

 C completely ; but this does not interfere with the use of the 

 resistance-operator for other purposes. 



* Especially Part III., and after, "On the Self-induction of Wires," 

 Phil. Mag. Oct. 1886 and after. Also 'Electrician/ Dec. 1884. 



