266 BELL SYSTEM TECHNICAL JOURNAL 



The electrical problem of the free oscillations of a resistanceless 

 network is formally the same as the dsnamical problem of the small 

 oscillations of a system about a position of equilibrium with no fric- 

 tional forces acting. The proof of formula (1) may be derived from 

 the treatment of this dynamical problem as gi\cn, for example, by 

 Routh." 



In any network the driving-point imix'dance in the qth mesh, Sq, 

 is equal to the ratio A/Ag, where -1 is the determinant* of the net- 

 work and Aq the principal minor of this determinant obtained by 

 striking out the ^th row and the 5th column. The determinant of a 

 network has the element Zjk in the jth row and ^th colunm, Zj^ 

 being the mutual impedance between meshes j and k (self-impedance 

 v:henj = k), the dclcrniinaiil including )i in<lc[H-n(lcnt nieshes of the 

 network. 



Hence the determinani .1 lias ilic element Zjk = iLjkp+{iCjkp) ', 

 where Ljk is the total inductance and Cjk the total capacity common 

 to the meshes 7 and k. Upon taking the factor («/>)"' from each row 

 and substituting —p-=x, the expression for A may be put in the 

 form A ={ip)~"D, where D is a determinani \\\\h LjkX-\-l/Cjk as the 

 element in the jth row and the A'tli column. This is of exactly the 

 same form as the delcrniinant giwn \ty Roiith '■' for the solution of 

 the dynamical problem; it is pro\ed there that this determinant, 

 regarded as a poKnomial, has 11 negative real roots which are separ- 

 ated by the «— 1 negatixe tea! roots of e\ery lir.si prim-i|)al niinur 

 of the determinani. 



Hence, we may write D = £(.vi+.v)(.V3+.v) . . . (.T2„_i+.y), where 



Xi, X3 Xin-i are all positive and arranged in increasing order of 



magnitude, and where E is also positive since D must be posit i\e for 

 x = 0. The determinant D^ may be expressed in similar manner since 

 it is of tile same form as 1) but of lnwii- (uder. 



' K. J. Routh, ".Advanced Rigid nynaiiiics," si.\lh edition, 1905, pages 44-55. 

 In the notation of the dynamical proljlem as presented here, the coeflicients A,,, 

 correspond to the inductances, l/C,^ to the capacities, p (ilir) to the frequency, 

 and 8', <t>', etc., to the branch currents in the electrical proMeni. 



A complete proof of formula (1) has been worked out for tlie electrical problem, 

 without depending in any way upon the solution of the corresponding dynamical 

 problem. This proof has not been published here in view of the great sim))litication 

 made by using the results already worked out for the dynamical problem. 



•A complete discussion of the solution of networks by means of determinants 

 has been given by G. A. Campbell, Transiictions of the A. I. E. E., 30, 1911, pages 

 873-909. 



» The determinant given by Koulli (Inc. cit., p. 40) has the element A,i,p"--^-C,i,. 



