36 Mr. L. C. Jackson on the Theory of the 



The Differential Equations and their Solution. 



Let us now consider the oscillations possible in three 

 coupled electric circuits in which the inductances, capacities, 

 and resistances are L l3 L 2 , L 3 , C l5 C 2} C s , R x , R 2 , II3 re_ 

 spectively, and which are inductively coupled through the 

 mutual inductances M 12 , M 23 , M 31 . 



The equations of motion of the circuits may be written : 



ei — I P J- T dil M d ^ M di * 



-= 22 K 2 + L 2 ^-M 23 --M 21 ^, 



n-^ 3 + L 3 ^~M 31 ^-M 32 ^ 



in which £ 1? <? 2jJ <? 3 are the charges on the three condensers in 

 circuits 1, 2, and 3 respectively at any instant ; i l9 i 2 , i 3 are 

 the corresponding currents, and also 



M 12 = M 21 , M 23 = M 32 , M 13 =M 31 . 



Since the current i is the time rate of decrease of charge e y 

 we have 



de x 

 It 





de c> 



It 



dez 

 It 





(1) 



Using equations (1), the above equations o£ motion may 

 be written in the form : 



,,|jp*i,T n d " ei M n d * e » 



M 13 C 1 g=0, 



-BA==f+LtQ,^-M»C.^ 



" 2 A< 



-, <P«, 



-M 21 O 2 ^=0 ; 



, i> n de* T _ d 2 e, 



M»Q 



eft 2 



-M 8 A$ = o. 



(2). 



