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



Special Cases. 



Let us now apply the theory developed above to various 

 special cases of the configurations of the coupled systems. 



Case I. 



We will now obtain the solution of equation (9) for the 

 following case. Let the oscillations be undamped, this case 

 resulting when the resistances in the circuits are negligibly 

 small either actually or virtually through the use of some 

 means such as the thermionic valve ; also let circuits 2 and 3 

 be identical in all respects, and let the couplings between 

 circuits 1 and 2 and 1 and 3 be equal. 



For this special case we may write 



R t = E 2 = R 3 = 0, 

 which gives 



q = r = s = k\ = k 2 = A' 3 = 0, 



m = n and a = <y. 



The solution of equation (9) can be more easily found 

 if instead of using the equation itself we use the partially 

 expanded form of the determinant from which it is 

 obtained. 



Eliminating the damping from the determinant and 

 writing co for p y we obtain 



ay 2 -l 2 {(o> 2 - mY-^co'} -2aW{/3co 2 + {co 2 -m 2 )} = 0, 

 which can be re-written 



(o) 2 (l + /3)-m 2 ){o) 4 (l-/3-2a 2 ) 



-o> 2 (Z 2 (l-0)+™ 2 ) + Z 2 m 2 } = 0. 

 From this we see that the three roots are 



2= Z 2 (l-ft)+ ? » a+ V / (/^l-/3)-f-m 2 ) 2 -4(l-^-2a 2 )/ 2 m 2 "> 

 Wl ~ 2(l-/3-2a 2 ) 



9 m 



0>3 = 



l 2 (l-0) + m*- \/{P(l-fi) + m 2 ) 2 -Ml-l3-2ot 2 )l 2 

 2 (l-/3_2a 2 ) 



2 



>(18) 



1 + /3 



For the oscillations to occur these roots must be real, 

 the condition for this being, since 1 + /3 cannot be negative, 



(P(l-£) +m a / -4(1-/3- 2a 2 )Z 2 m 2 > 0. 



