176 Mr. 0. Heaviside on the 



If the lines are representable by resistance, self-induction, 

 electrostatic capacity, and leakage conductance (R, L, S, K 

 of Parts IV. and V., per unit lengths), one line will be a 

 reduced copy of the other if, when R and L in the first line 

 are n times those in the second, 8 and K in the second are 

 n times those in the first, in similar parts. 



After these general remarks, and preliminary to the con- 

 sideration of the quadrilateral, let us briefly consider the 

 general theory of the conjugacy of a pair of conductors in a 

 connected system, when an impressed force in either can cause 

 no current in the other, either transient or permanent. The 

 direct way is to seek the full differential equation of the cur- 

 rent in either, when under the influence of impressed force in 

 the other alone. Let V = ZC be the differential equation of 

 any one branch, being the current in it, V the fall of 

 potential in the direction of C, and Z the differential operator 

 concerned, according to the notation of Parts III., IV., and 

 Y. If there be impressed force e in the branch, it becomes 

 g + V = ZC. We have 2V = in any circuit, by the potential 

 property; therefore 2<? = 2ZC in any circuit. Also the cur- 

 rents are connected by conditions of continuity at the junctions. 

 These, together with the former circuit equations, lead us to a 

 set of equations : — 



FC 2 =/ 21 e 1+ / 22 e 2 + ...V .... (lc) 



: :} 



C 1? C 2 , . . ., being the currents, and e ly e 2 , . . . the impressed 

 forces in branches 1, 2, &c. ; F being common to all, and it 

 and the/'s being differential operators. We arrive at similar 

 equations when the differential equation of a branch is not 

 merely between the Y and C of that branch, but between 

 those of many branches ; for instance when 



Y 1 = Z 11 1 + Z 12 2 + ... . • . . (2c) 



is the form of the differential equation of branch 1. 



Now let there be impressed force e in one branch only, and 

 be the current in a second, dropping the numbers as no 

 longer necessary. We then have 



FC=/(.) (3c) 



Conjugacy is therefore secured by /(e) =0, making C inde- 

 pendent of e. Therefore f(e) = is the complex condition of 

 conjugacy. If, for example, 



f(e) = a e + a x e + a 2 e + . . ., .... (4c) 

 where the a's are constants, functions of the electrical con- 



