178 



Mr. 0. Heaviside on the 



a 2 



o, 



(10c) 

 (lie) 



(12c) 



and therefore 



Thus, if a 2 = also, we have 



1 dt(*Gdi = 0. 



Jo Jo 



Similarly, if a 3 = also, then 



j d* \ dt\ Gdt = 



Jo Jo Jo 



and so on. The physical interpretation of a =0 and ^ = is 

 obvious, but after that it is less easy. 



If F contain inverse powers of p, the steady current may 

 be zero. But in spite of that, it will be found that to secure 

 perfect conjugacy for transient currents, we must have a true 

 resistance -balance, or that relation amongst the resistances 

 which would make the steady current zero, if we were to 

 allow the possibility of a steady current by changing the 

 value of other electrical quantities concerned. I will give an 

 example of this later. 



I have elsewhere* pointed out these properties of the func- 

 tion F, in the case where there is no mutual induction, or 

 Y = Z0 is the form of the differential equation of a branch. 

 Let n points be united by ^n (n — 1) conductors, whose con- 



ductances are K 

 numbered 1, 2, &c. 



12) 



K 13 , &c, it being the 

 Then the determinant 



K-n, K 12 , . . ., K ln 



-K-21? &$2) • • •) ^2n 



points that are 



l^w.l* -K-,,,2' 



-nl) 



•) ^-nn 



is zero, and its first minors are numerically equal, if any K 

 with equal double suffixes be the negative of the sum of the 

 real K's in the same row or columnf . Remove the last row 

 and column, and call the determinant that is left F. It is the 

 F required, and is the characteristic function of the combina- 

 tion, expressed in terms of the conductances. If every branch 

 have self-induction, so that ~R + h(d/dt) takes the place of 

 K _1 , then F = is the differential equation of the combination, 

 without impressed forces, and F = is always the differential 

 equation subject to the condition of no mutual induction. In 



* 'Electrician/ Dec. 20, 1884, p. 106. 

 f As in Maxwell, vol. i. art. 280. 



