500 Mr. 0. Heaviside on Resistance 



magnetic forces in those parts of the system where the per- 

 mittivity and the inductivity are finite, or are reckoned finite 

 for the purposes of calculation. Thus conductors, if they be 

 not also dielectric, have only to be considered as regards the 

 magnetic force, whilst in a dielectric we must consider both 

 the electric and the magnetic force. [The failure of Max- 

 well's general equations of propagation arises from the impos- 

 sibility of expressing the electric energy in terms of his 

 potential function. The variables should always be capable 

 of expressing the energy.] Now the internal connexions of a 

 system determine what ratios the variables chosen should 

 bear to one another in passing from place to place in order 

 that the resultant system should be normal ; and a constant 

 multiplier will fix the size of the normal system. Thus, sup- 

 posing u and w are the normal functions of potential-difference 

 and current, which are in most problems the most practical 

 variables, the state of the whole system at time t will be 

 represented by 



V=2Au€** C = 2AiW; . . . (46) 



V being the real potential-difference at a place where the 

 corresponding normal potential-difference is u, and C the 

 real current where the normal current is w, the summation 

 extending over all the p roots of the characteristic equation. 

 The size of the systems, settled by the A's (one for each p) 

 are to be found by the conjugate property of the vanishing of 

 the mutual energy-difference of any pair of p systems, applied 

 to the initial distributions of V and 0. 



17. To find the effect of impressed force is a frequently 

 recurring problem in practical applications ; and here the 

 resistance-operator is specially useful, giving a general solu- 

 tion of great simplicity. Thus, suppose we insert a steady 

 impressed force e at a place where the resistance-operator is 

 Z, producing e = ZC thereafter. Find C in terms of e and Z. 

 The following demonstration appears quite comprehensive. 

 Convert the problem into a case of subsidence first, by sub- 

 stituting a condenser of permittance S, and initial charge Stf, 

 for the impressed force. By making 8 infinite later we arrive 

 at the effect of the steady e. In getting the subsidence solu- 

 tion we have only to deal with the energy of the condenser, 

 so that a knowledge of the internal connexions of the system 

 is quite superfluous. 



The resistance-operator of the condenser being (Sp)" 1 , that 

 of the combination, when we use the condenser, is Z v where 



Z 1 = (Sp)" 1 + Z (47) 



Let Y and C be the potential-difference and the current, 



