394 PELL SYSTF.M TECIIMC.-II. JOVRMAI. 



in accorclaiu'c with (3), tht- iMrrcspnuliiii; (iisirilnitiim-^ nf network 

 currents // . . . . /„' and /i" .... /„" an: yiwii 1)\ 



// = 2-'l.,*lV, j^],2...>:. (4) 



//'=2--l>A.n.". (5) 



/r = l 



Now l<irm the product sum .^1/7/: !)>■ nit'aii> of (4) it is easy to 

 show that, since Ajk = Akj, 



Since this is symmetrical in the two sets of applied forces V\' .... V„' 

 and Ki" .... Vn", it follows at once that 



which proves the theorem. 



Now if we analyze the foregoing proof it is seen to dejieiid on ihe 

 assumption, first that the network can be described in terms of a set 

 of simultaneous equations with constant coefficients, and secondh" 

 on the reciprocal relation in the coefficients, Zjk = 'Lkj- In other 

 words, it is assumed that the currents flow in linear, invariable cir- 

 cuits, and that the system is what is called quasi-stationary.- What 

 this means is that the network may be treated as a dynamical system 

 defined by n coordinates, the n currents l\ . . . .l„ being the \-eloci- 

 ties of the n coordinates. More precisely stated, the underlying 

 assumption is that the magnetic energy, the electric energy, and the 

 dissipation function can be expressed as homogeneous quadratic 

 funclioiis of ihi- following form 



H'=A2 2-^*ftC^*. lj = ddtQj, 

 and 



where the coetlicii-nls L,k. S,k. Rjk are I'onslants. Subject to tiiese 

 assumptions, which, it may be remarked, underlie the whole of electric 

 circuit thcor>', the direct application of Lagrange's etiuations to the 

 (juadralic functions T, W, D leads at once to the circuit equations (1) 

 and the recijirocal relation Zjk = Zkj. This is merely a very brief outline 

 of Maxwell's dynamical theory of quasi-stationary systems or networks. 

 * See Theoric der Electrizilat, .Abraham u. Koppl, Vol. I, p. 254. 



