Self-induction of Wires. 337 



tion should be linear, so that Z may be a function of p. If, 

 for example, (dC/dt)' 2 occurred, we could not do it. 



Now this combination must necessarily be joined on to an- 

 other, however elementary, to make a complete system, unless 

 V is to be zero always. The complete system, without im- 

 pressed forces in it, has its proper normal modes of subsidence, 

 corresponding to definite values of p. Consequently, by (96), 



U 13 -T 12 =(V ? C 1 -ViC 2 )-=-(p 1 -. a ), . . (99) 



if Vi, Ci belong to p l9 and V 2 , C 2 to p 2 , whilst the left member 

 refers to the combination given by V = ZC. Or 



u 12 -t 12 = ca ( Ji - £) i- fo-po = ca %=2> (ioo) 



\^1 ^2/ P'2 — Pi 



and the value of 2(U— T) in a single normal system is 



2(U-T)=v|- C f ._*!;._(*!.. (101) 



In a similar manner we can write down the energy- 

 differences for the complementary combination, whose equation 

 is, say, Y = YC ; remembering that — YC is the energy 

 entering it per second, we get 



CiCo— and C 2 -r- respectively. 



Pi— Pa dp r J 



By addition, the complete U 12 — T 12 is 



C 1 C 2 Yl ~ Y2 ~ Zl + Z2 =Q = C 1 C 2 ^^; . (102) 



Pl~p2 P1-P2 ' 



and the complete 2(U— T) is 



C 2 |(Y-Z),orC^, . . . (103) 



where $ = 0, or Y — Z = 0, is the determinantal equation of the 

 complete system (both combinations which join on at a and b, 

 where Y and C are reckoned), expressed in such a form that 

 every term in <f> is of the dimensions of a resistance. 



If the complete system depends only upon a finite number 

 of variables, it is clear that the number of independent normal 

 systems is also finite, and there is no difficulty whatever in 

 understanding how any possible initial state is decomposable 

 into the finite number of normal states ; nor is any proof 

 needed that it is possible to do it. The constant Ai, fixing 



