CIHCl'IT THEORY ANO Ol'i.KA itOXAl. CAI.CVLVS 695 



Then if /i . . . /, is a solution of (1), Ii\ Ix, . . . /, + /-■', is also a 

 solution. 



To derive the solution of the complementary system of eiiiiatiuns 

 (.14), assume that a solution exists of the form 



// = -//<'^' 0=1,2..«) 



so that d/dl = \ andj dt=\/\. Substitute in ecjuations (It) and 

 eancel out the common factor e^'. Then we have 



(15) 



Zn,{\)Jv'-\- +Znn{\)Jn'=0. 



This is a system of n homogeneous equations in the unknown quan- 

 tities Ji, . . Jh- The condition that a finite solution shall exist is 

 that, in accordance with a well known principle of the theory of 

 equations, the determinant of the coefficients shall \anish. That is, 



Zn(X) .... Zu, (X) ; 



D(\) = 



Z„i{\) . . . .Znn (X) 



(16) 



Consequently the possible values of X must be such that this equation 

 is satisfied. In other words, X must be a root of the equation D{\) = o. 

 Let these roots be denoted by Xi, Xj . . \m. Then, assigning to X any 

 one of these values, we can determine the ratio J/ /Jk from any (« — I) 

 of the equations. That is to say, if we take 



/.' = c/''+C,f^'+ .... +C„e^"''. (17) 



substitution in any (n— 1) of the equations determines /j', . . In ■ 

 The m constants C\, . . Cm are so far, however, entirely arbitrary, 

 and are at our disposal to satisfy imposed boundary conditions. 



This introduces us to the idea of boundary conditions which is of the 

 greatest importance in circuit theory. In physical language the 

 boundary conditions denote the state of the system when the electro- 

 motive force is applied or when any change in the circuit constants 

 occurs. The number of independent boundary conditions which 

 can, in general, be satisfied is equal to the number of roots of the 

 equation D{\)=o. Evidently, therefore, it is ph>sically impossible 

 to impose more boundary conditions than this. On the other hand, 

 if this number of boundary conditions is not specified, the complete 

 solution is indeterminate : That is to say, the problem is not correctly 

 set. As an example of boundary conditions, we may specify that the 



