525-528] Pair of Circuits 457 



now become identical. We no longer can deduce the relations A^ = Ar' 2 = 0, 

 but have only the single initial conditions 



But by supposing equations (455) and (456) replaced by equations (455) 

 and (458) we have only one differential coefficient and therefore only one 

 constant of integration in the solution, and this can be determined from 

 the one initial condition expressed by equation (458). 



528. Let us, for instance, consider the definite problem discussed (for the 

 general case) in 520. Circuit 2 contains no battery so that E. 2 = 0, and at 

 time t = circuit 1 is suddenly closed, so that the electromotive force E l 

 comes into play in the first circuit. The initial currents are given by 



(from equation (458)), ............ Z^ + l/V^O .............................. (459), 



(from equation (457)), ......... ME, = RMi,- SLi ........................ (460), 



i, i, ME, ME, 



so that 



M~~ - L RM* + SL* ~ L(RN+ SL) 



Thus finite currents come into existence at once, but the system of 

 currents is one of zero energy, since equation (459) is satisfied. To find the 



subsequent changes, we multiply equation (455) by -~ and equation (456) by 



-~- (putting E Z = Q), and find on addition 

 LE, (L N\ d , 



of which the solution, subject to the initial condition Li, + Mi 2 = 0, is 



TF / BS 



Li, + Mi, = ^ (1 - 



This and equation (460) determine the currents at any time. 



These results can of course be deduced also by examining the limiting 

 form assumed by the solution of 520, when LN - M* vanishes. 



The problem of the breaking of a circuit, discussed in 521, can be 

 examined in a similar way in the special case in which LN M 2 = 0. 



REFERENCES. 



J. J. THOMSON. Elements of the Mathematical Theory of Electricity and Magnetism, 

 Chap. XL 



MAXWELL. Electricity and Magnetism, Part iv, Chap. in. 



