THE DYNAMICS OF SIMPLE CIRCUITS 27 



twice. There result then seven equations, from which the six quan- 

 tities e and j and their derivatives can be eliminated : 



(12) 



a" + [a + b - (A - B)E(£ + a)]a' 

 - (bA - aB)E(§ + a) + aba = 0. 

 Equilibrium occurs for e , / satisfying 



E(£ + e - j) = ae/A = bj/B , (13) 



and hence for v = e — j satisfying 



(A/a-B/b)E($ + ,r) =a. (14) 



The action of the circuit is somewhat different for the two pos- 

 sible signs of the coefficient of E . If this is positive, this equation 

 has the same form as equation (5). If 



A/a -B/b^l, 



there is always a single non-negative root <r of equation (14). If the 

 relation fails there is at least one, and there may be three. Suppose 

 cr is a positive root, the largest if there are more than one. Let 



(15) 



E (£ + a) = e n + eiX + e 2 x 2 • • • 



where 



e = a /(A/a -B/b). (16) 



Then equation (12) can be written 



x" + [>+&- (A - B)e 1 ] x 



(17) 

 + lab - (bA - aB)e l ] x + ~- = 



with terms of second and higher degree in x omitted. 



Now at the value o- considered, the slope of the left member of 

 equation (14) must be less than one, and this, in view of the expan- 

 sion (15), means that the coefficient of x in equation (17) is positive. 

 Hence the characteristic roots of the linearized equation (17) are 

 either complex or else real and of the same sign ; if, further, 



a + b> (A -B)e lf (18) 



the real parts are both negative ; and if, finally, 



(\M - \ r B) 2 e 1 <a-b< (V~3 + yjB) 2 e x , (19) 



the roots are complex. Hence if the relation (18) is satisfied, the 

 equilibrium a n is stable, and if relation (19) also holds, the approach 

 to the equilibrium value fluctuates with a frequency v satisfying 



