180 



Mr. 0. Heaviside on the 



conductors unite 5 points, the currents in at least 6 conductors 

 must be given, and no four of them should meet at one point. 

 The remaining conductors are n — 1 in number, or one less 

 than the number of points, and n— 1 is the degree of the 

 characteristic function in terms of the conductances. Now 

 put F = in terms of the resistances, by multiplying by the 

 product of all the resistances. It is then made of degree 

 i(n — l)(w — 2) in terms of the resistances, which is the num- 

 ber of current freedoms. If n = 4, the degree is the same, 

 viz. three, whether in terms of conductances or resistances ; 

 but if n = 5, it is of the sixth degree in terms of resistances 

 and only of the fourth in terms of the conductances ; and if 

 n=6, it is of the tenth degree in terms of the resistances, but 

 only of the fifth in terms of the conductances, and so on ; so 

 that F becomes enormously more complex in terms of resist- 

 ances than conductances. 



When every branch has self-induction, Z = R-f-Lp, and the 

 degree of p in F = is the number of freedoms, so that there 

 are n — 1 fewer roots than the number of branches. It is the 

 same when there is mutual induction. The missing roots 

 belong to terms in the solutions for subsidence from an arbi- 

 trary initial state which instantaneously vanish, producing a 

 jump from the initial state to another,, which subsides in time. 



On the other hand, if every branch (without self-induction) 

 is shunted by a condenser of capacity S 1? S 2 , &c, K becomes 

 K + Sp, so that the degree of p in F = is the same as that 

 of K, or i(w— l)(w— 2) fewer than the number of con- 

 densers *. 



Coming next to the Wheatstone quadrilateral self-induction 

 balance, let there be six conductors, 1, 2, &c, uniting the four 

 points A, B 1? B 2 , C in the figure. ABiC and AB 2 C are the 

 lines referred to in the beginning, 

 and L the inductance of a 

 branch in which the current is 

 C, reckoned positive in the 

 direction of the arrow, and the 

 fall of potential Y in the same 

 direction ; thus R 1? L x , V l5 C^ A 

 for the first branch. The six 

 branches may be conjugate in 

 pairs, thus : 1 and 4, or 2 and 

 3, or 5 and 6. In the follow- 

 ing 5 and 6 are selected always, 

 the battery or other source 

 being in 6, and the telephone 



Let R be the resistance 



Electrician/ Jan. 1, 1886, p. 147. 



