690 LiEl.I. SYSTI-.M TECIIXICAL JOIRX.IL 



in wliicli these properlies no longer obtain. The solution, ho\ve\er, 

 for the restricted system of linear equations with constant coefficients 

 is fundamental and its solution can be extended to important jirob- 

 lems in\()lving non-linear relations and variable coefficients. Tiu-se 

 extensions will be taken up briefly in a later chapter. 



Another important property is the reciprocal relation among the 

 coefficients; that is Ljk=Lkj: Rjk = Rkj, and Cjk=Ckj. It is easily 

 shown that these reciprocal relations mean that there are no con- 

 cealed sources or sinks of energy'. Again important cases exist where 

 the reciprocal relations do not hold. Such exceptions, however, 

 while of physical interest do not affect the mathematical methods 

 of solution, to which the reciprocal relation is not essential. 



Returning to equation (1) we shall now derive the equation of 

 activity. MuliipK- tlio first equation by /i, the second by /;, etc. and 

 add : we get 



The riglu liand >i(lL' is the rate at which the applied forces are siii^jilying 

 energy to the network. The first term on thi' k-ft is ilic rale of in- 

 crease of the magnetic energy 



-j-y^^^z.,,./,/,, 



while the second term is the rate of increase of the electric energy 



Tlie last term, "S^ Rjk Ij Ik, is the rate at wliirli electromagnetic 

 energy is being converted into heat in the network. Consequently 

 in the electrical network, the magnetic energy is a homogeneous 

 quadratic function of the currents, the electric energy is a homogene- 

 ous quadratic function of the charges, and the rate of dissipation 

 is a homogeneous quadratic function of the currents. In Maxwell's 

 dynamical theory of electrical networks, these relations were written 

 down at the start and the circuit equations then derived by an ap- 

 plication of Lagrange's dynamic ecjuations to the hfimogencous <|u.id- 

 ratir funrtifins. 



Returning to ec|uations (1), we observe that, due to the presence of 

 the integral sign, they are integro-diflferential equations. They are, 



