4^ 



692 



BELL SYSTEAf TECHSICAL JOURNAL 



ihe exponential solution so simple, since we can immediately pass 

 from differential ec|uations to algebraic equations. In these algebraic 

 equations, « in number, there are n unknown quantities J\, . . Jn- 

 These can therefore all be uniquely determined. We thus see that 

 the assumed form of solution is possible. 



The notation of equations (4) may be protiiabh- simplified as fol- 

 lows: write 



\Ljk + Rjk + 1 /XC> = 'j*(X) = zjk 



;ind wc ha\e 



221-^1 +S22-/-2 + . .+=2./-.=0, 



(5) 



S„.-/l+S..2-/2+. .+=«,./,. =0. 



The solution of this s\-stcm of ecjuations is 



■^'~ D{\) ''" D ' 



md 1 r c 



ivhere D is the determinant of the cocfticicnts, 



-U -12 

 221 S22 

 231 S32 



(6) 



(7) 



S,.l In 2 



;ind Mji is the cofactor, or minor with proper sign, of the jth column 

 and first row. 



I shall not attempt to discuss the theory of determinants on which 

 this solution is based.' We may note, however, one important 

 property. Since Zjk = Zkj, Mjk = Mkj. From this the Reciprocal 

 Theorem follows immediately. This may be stated as follows: 



If a force Fe^' is applied in the jth mesh, or branch, of the net- 

 rtiirk, the current in the k\h nu'sli, or branch, is by tiie foreiioing 



' n /•'■ • 



Now apply the same force in the A'lii mesh, or braiuii, then the cur- 

 rent in the jth mesli is 



l^tc ■ 



' Kor a remarkably hhk [>r .mil cjniplt'te discussion of the exponential snlution 

 by aid of the thcorv of delerniinants, sec Cisoidal Oscillations. 1 rans. .X. I. I-.. K., 

 1911, by G. .\. Canipbcll. 



