Self-induction of Wires. 65 



they are similar when 



Z,/Z 2 = ra, any numeric (4rf) 



Here Z is the symbol of the generalized resistance of a 



• • • i 



system between its terminals, when it is, save for its terminal 



connexions, independent of all other systems ; a condition 

 which is necessary to allow of the form V = ZC being the full 

 expression of the relation between V and C, Z being a func- 

 tion of constants and of p.p 2 , p s , &c, and p being d/dt. To 

 ensure the possession of the property (4c/), we require first of 

 all that one system should have the same arrangement as the 

 other, as a coil for a coil, a condenser for a condenser, or equi- 

 valence (as, for instance, by two condensers in sequence being 

 equivalent to one) ; and, next, that every resistance and in- 

 ductance in the first system be n times the corresponding 

 resistance and inductance in the second system, and every 

 capacity (electrostatic) in the second system be n times the 

 corresponding one in the first. 



Then, if the two systems be joined in parallel, and exposed 

 to the same external impressed force at the terminals, the 

 potentials and electromotive forces will be equal in corre- 

 sponding parts, whilst the current in any part of the second 

 system will be n times that in the corresponding part of the 

 first. Also the electric energy, the magnetic energy, the dis- 

 sipativity, and the energy-current in any part of the second 

 system are n times those in the corresponding part of the first. 



The induction-balance got by joining together correspond- 

 ing points through a telephone is, of course, far more general 

 than the Christie balance, limited to four branches, each sub- 

 ject to V = ZC ; at the same time, however, it is less general 

 than the conditions which result when the full differential 

 equation is worked out*. 



By the above, any number of similar systems may be joined 

 in parallel, having then equal electromotive forces, and their 

 currents in the ratio of the conductances. They will behave 

 as a single similar system, the conductance of any part of 

 which is the sum of the conductances of the corresponding 

 parts in the real systems ; and similarly for the capacities and 

 for the reciprocals of the inductances. If, on the other hand, 

 they be put in sequence, the resultant Z is the sum of the 

 separate Z's, the current in all is the same, and the electro- 

 motive forces are proportional to the resistances. 



* This general property is, it will be seen, of great value in enabling 

 us to avoid useless and lengthy mathematical investigations. In the 

 • ' Electrician * fur May 27, 1887, 1 have shown how to apply it to the at 

 first sight impossible feat of balancing iron against copper. 



Phil. Mag. S. 5. Vol. 24. No. 146. July 1887. F 



