498 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XII. 



damping of any particular simple oscillation are the same for all 

 the coordinates, and the n factors of the amplitudes and the n 

 phases are to be determined from the initial values of the q's and 

 of their first time derivatives. 



We will now consider the case of forced vibrations. 



On account of the linearity of the equations, if we find a solu- 

 tion q r (l) for a particular set of values E g (l] of the right-hand 

 members of our equations (7), and a second solution q 8 ( ^ for a 

 second set E^\ then the sum q r (l) + q r (2} will be the solution when the 

 right-hand members are E s (l) + E. We shall, therefore, consider 

 the effect of each impressed force by itself. Suppose first then 

 that in each circuit there is impressed a harmonic electromotive 

 force, E s cos cot, all of the same period. Then we have the equations 

 of which the 5th is 



(20) 



Assuming q r = a g e i(at these reduce to 



(- M u a>* + R n ico + p u ) Oj + ... + (- M ln co 2 + R ln ico + p ln ) a n 



= E 



n 



a set of linear equations to determine the a's. 



If we call the determinant of equation (10), D (X), and D rs (X) 

 the minor of the element of the rth column and 5th row, we have 

 as the solution of (21), 



<-> --^^- 



Since D (X) = is the determinantal equation for the free vibration, 

 whose roots are Xj, X 2 ... \ Zn , we have 



(23) D (X) = C (\ - \) (\ - X 2 ) ... (X - X 2w ) = CU g (X - X g ). 

 Accordingly the denominator D (ico) is 



(24) D (ia>) = CU S (ico - X s ) = CU S {- 



The minors D rs (iw) are rational integral functions of ia), and the 

 numerators are therefore complex quantities, which reduce to real 



