CASE OF TWO CIRCUITS. 517 



540. CASE OF Two CIRCUITS. Let us consider two adjacent 

 circuits C and C', whose relative position is fixed, and which contain 

 electromotive forces ; we shall represent by E, R, and L the electro- 

 motive force, the resistance, and the coefficient of self-induction of 

 the first circuit ; by the same quantities accentuated, the analogous 

 quantities of the second ; and by M the coefficient of mutual 

 induction. All the coefficients being supposed constant, we shall 

 have the two simultaneous equations 



M + L + RI-E = 0, 



dt dt 

 (27) dl dl , 



M + L' + RT-E' = 0, 

 dt dt 



the solution of which is of the form 



RI -E= 

 ' 



The coefficients A, B, A', B' are constants to be determined from the 

 conditions in respect to the limits. 



Expressing the condition that these values of the strength satisfy 

 the differential equations (27), whatever be the time, we find that the 

 exponents p and p are roots of the equation of the second degree. 



/ M 2 \ /R R'\ RR' 



(29) ( i -- U2 + / _ + _ \p+ -- = o. 



\ LL'/ \L L'/^ LL' 



These roots are always real ; again they must be negative, seeing 

 that, under no circumstances, can the strength increase indefinitely 

 with the time. Hence we must have LL/>M 2 , which moreover is 

 evident (519); we could only have LL' = M 2 if the two circuits 

 coincided. 



It follows also from this condition that if the coefficient of self- 

 induction L of a wire is very small, the coefficient of mutual 

 induction M of this wire with any other wire is also very small. 



541. Consider the case in which E' = 0, that is to say in which 

 the second circuit contains no electromotive force. 



If we only wish to determine the quantities of electricity m and 

 m' which traverse the two circuits in time /, we may calculate the 



