241] INDUCTION OF CUKRENTS. 495 



the as, however, the determinant of the coefficients must vanish, 

 namely 



(10) 



lf n X 2 



0. 



nl \* + R nl \ + p n , ...... M nn \ 2 + R nn \ + p n 



This is an equation of order 2n in X, from which the odd 

 powers are absent if F= 0. We shall denote its roots by 



Xi, X-2, ...... ^m- 



If we multiply the rth equation (9) by a r , and take the sum 

 for all r's, we obtain 



( 1 1 ) \^ r ^ s M rs a r a s + \^r^ s R rs a r a s + % r % s p rs a r a s = 0. 



The double sum by which X 2 is multiplied is the value of the 

 function 2T when for every q s ' is substituted a s . We shall denote 

 this by 2T (a). Similarly the coefficient of X is 2F (a) and the term 

 independent of X is 2TT(a). But by the fundamental property of 

 the three functions, each must be positive. The equation (i i), 



KT(a) + \F(a) +W(a) = 0, 



shows us at once that X can not be real and positive, for that 

 would involve the sum of three positive terms being equal to zero. 



Secondly, if F = 0, that is, if the resistance of every wire is 

 zero, 



and X is a pure imaginary. In this case &* and e~ M are trigono- 

 metric functions, representing an undamped oscillation of the 

 same period for all the parameters q. 



Thirdly, if F is large enough, X can be real and negative. In 

 this case each parameter q gradually dies away to zero, the relax- 

 ation time being the same for all. This corresponds to Case I of 

 239. 



Fourthly, if either W or T is zero, instead of a pair of roots we 

 have a single one, which is real and negative, the cases correspond- 

 ing respectively to 237 or 207. 



Fifthly, in other cases, that is when neither T, F, nor W 

 vanish, and F is not too large, X is complex. We shall prove that 

 then the real part of X is negative. 



