474 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XII. 



These equations can be satisfied for values of A and B differing 

 from zero only if the determinant of the coefficients, 



M\ , 2 X + R 

 vanishes. But this being expanded gives us the equation 

 (7) (L,L 2 - M*) X 2 + (R,L, + R.L,) X + R,R 2 = 0, 



a quadratic to determine X. If we call its roots \ and X 2 , we have 



, ,, 2 , 2 ,, - AT 2 ) 



x = 



Both roots are real, for we can write the quantity under the 

 radical sign 



both terms of which are positive. Both roots are also negative, 

 for since the electrokinetic energy 



is intrinsically positive, we must have 



L,L, - M 2 > 0. 



Having found the value of X either of the equations (6) will give 

 us the ratio of the constants A, B. If we choose the value X x 

 the first equation gives 



(a\ ?!. - -^Ai + -Ri 



Ai M\, 



If we choose the value X 2 we obtain a different ratio 



(10) B ? ___L l \ 2 + R l 



A,' M\ 2 



The theory of linear differential equations shows that the sum of 

 particular solutions is a solution, and that the general solution is 

 given by 



where the constants A lt B lt A a , B z are connected by the equations 

 (9) and (10). We may now determine the absolute values of these 



