517-520] Pair of Circuits 451 



during the interval from = to t = r the values of -j (Li^ + Mi 2 ) and of 



-T- (Mi^ + Ni 2 ) are finite, so that when r is infinitesimal, the changes in 

 (it 



Li^ + Mi 2 and Mi 4- Ni 2 must vanish. Thus we must have 



ZA^ + 3/A?' 2 = 0, 



Except in the special case in which LN M~ = (a case of importance, 

 which will be considered later), these equations can be satisfied only by 

 Ai\ = Ai 2 = 0. Thus the currents remain unaltered by suddenly making a 

 circuit, and the change in the currents is gradual and not instantaneous. 



520. Suppose, for instance, that before the instant = circuit 2 is 

 closed but contains no battery, while circuit 1, containing a battery, is broken. 

 Let circuit 1 be closed at the instant t 0, then the initial conditions are 

 that at time t = 0, i\ = i 2 = 0. The equations to be solved are 



d ^' " d ' ~ .(442), 



M i, 



The solution is known to be 



where A, A', B, B' are constants, and X, X 7 are the roots of 



(R - L\)(S - N\) - M*\* = 0, 

 or of RS-(RN+SL)\ + (LN-M*)\* = ............ (444). 



The energy of the currents, namely 



being positive for all values of ^ and i z , it follows that LN-M* is necessarily 

 positive. Since RS and RN + SL are also necessarily positive, we see that all 

 the coefficients in equation (442) are positive, so that the roots X, X' are both 

 positive. 



When =0, we must have 



(h\= Q = A + A' + 1 = ..................... (445), 



= ........................ (446), 



and in order that equation (443) may be satisfied at every instant, we must 

 have 



- N\) Ee~ Kt + (S- N\') E'e~^ = 0, 



292 



