I'Hinxci'iiiM iMi'i.DAScr. (>/■ nroMi.sii cificiiis wi 



TIk' ni-twiirk (liaKr.iin> ami all llu- fiirnHila-< for I lie I'.ili'iilatinii of 

 the fk-nu'ius ri'iD.iin iinrlianv;»'<l. 



S. MaIHI \I Mil Al 1'ROOI" 



The circuits treatetl in this iincstii;,iti(in arr .-.iH'cial c.iscs of tin- 

 Kt-niTal circuit which has ,iin iiuniln-r of ifriiiinals m connected in 

 pairs by »i(»i — I) 2 branches, each of which consists of a self-induct- 

 ance, a resistance, and a capacity in series, with mutual in<luctance 

 lietween each pair of branches. The only restrictions iniposed are 

 those inherent in all electrical circuits, namely, that the magnetic 

 energy, the dissipatit)n, and the electric energy are each positi\e for 

 any possible distribution of currents in the branches. Circuits with 

 any arrangement of elements in series or in parallel or in separated 

 meshes can be derived as limiting cases of this general circuit by 

 making a sufficient number of the inductances, resistances, and capaci- 

 ties either zero or infinite. 



This general circuit connecting in terminals or branch-points has 

 « = (m— I)(»i— 2) 2 degrees of freedom, that is, « independent 

 meshes. The discriminant " of the circuit is the determinant A 

 having the element Z;* in thejth row and ^th column, Zjk being the 

 mutual imjx'dance between meshes j and k (self-impedance when 

 j = k), the determinant including n independent meshes of the circuit. 



The driving-point impedance in the 9th mesh S^ is equal to the 

 ratio ,4 ,4,,,;, where .4,,,, is the cofactor of the element in the qth 

 row and qth column of the determinant .4. In general, the cofactor 

 of the product of the elements located at the intersection of rows 



j, q, s, . . . with columns k, r, t respectively, will lie denoted by 



Ajt^ St, . .. 



The determinant -4 for the general circuit described aboNe is of 

 order « with the element 



Z„ = iLjkp+Rjk+(.iC,kp) " (38) 



where L,k. Rjl-. and Cjk are the inductance, the resistance, and the 



capacity, respectively, common to the two meshes j and k. The 



inductance Ljk includes, therefore, the self-inductances of the branches 



common to the two meshes together with the mutual inductances 



connecting each branch of one mesh with each branch of the other 



mesh. The determinant is symmetrical, that is Z>* = Zt->, since 



Ljk = Lkj, Rjk = Rkj. and Cjk = Ckj. 



" A complete disrussion of the solution of circuit-s hy means of iletcnnlnants has 

 iM-en given by G. .A. Campliell, Inc. cit., pages 88.?-88<>. 



