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



values minus one and plus one respectively. We shall write our 

 three functions 



(5) 



where the M'B are linear combinations of the inductances of the 

 wires, the It's linear combinations of their resistances, and the ps 

 linear combinations of the coefficients of electrostatic potential of 

 the accumulators. The values of the coefficients of the three 

 functions are such that each of the functions is positive for all 

 possible choices of its variables. 



We may now apply Lagrange's equations for any parameter q s . 



d fdT\ dF dW 

 (6) -j-A*-. ' ) + 5-' + -^r- = ] s> 



dt \dq s J dq s dq s 



where E s is the total external electromotive-force around the 

 circuit. Performing the differentiations this becomes 



a linear differential equation of the second order with constant 

 coefficients. We have one such equation for each parameter q s . 



We shall first find the free oscillations, that is the solutions 

 with every E s = 0. As in the case of the simple examples of 237, 

 a particular solution may be obtained by assuming for every q s , 



(8) . 



where X is the same for all the qs. Inserting these values in (7) 

 we obtain 



+p u ) ox + ...... + (M ln \* + Rm^+Pm) a n = 0, 



+ ...... + (M 2n \* + R m \ + p) a n = 0, 



(9) ........................................................................ 



(M m \* + R nl \ +p nl ) a,+ ...... 4- (M nn \* + R nn \ +p nn ) a n = 0, 



a set of linear equations to determine the ratios of the a's when X is 

 known. If these are to be satisfied by other than zero values of 



