considered in connexion with the Dynamical Theory. 13 



confusion seems to have prevailed, as shown by the numerous 

 inquiries into the direction of the induced currents of high 

 orders. The currents, as a whole, at least after the first, can- 

 not properly be said to have any direction at all, as they in- 

 volve, when complete, no transfer of electricity in any direc- 

 tion. Nevertheless the positive and negative parts are not si- 

 milar ; and if they were, one must necessarily precede the other ; 

 so that in this way directional effects may be produced. The 

 magnetizing power, for instance, depends essentially on the initial 

 maximum magnitude of the induced current, and is probably 

 but little affected by the character of the diluted but compara- 

 tively long-continued remaining parts. This being understood, 

 the alternately opposite magnetizations observed by Henry in a 

 series of induced currents of high order, is an immediate con- 

 sequence of the dynamical theory. 



The circuits being denoted by the numbers 1, 2, 3, ... , let 

 the coefficient of mutual induction between 2 and 3 be denoted 

 by (2 3), and of self-induction of 2 by (2 2), and so on. The 

 result is only generally true when there is no mutual induction 

 except between immediate neighbours in the series ; and it will 

 therefore be supposed that 



(13), (14), (15). ..(34)... 



vanish, as indeed they practically would in the ordinary arrange- 

 ment of the experiment. The energy of the field is given by 



E=i(ii)*!+*(2 *J«!+*(3 &)*!+... 



+ (1 2)^2+ (2 S)x 2 x 3 + (3 4)^ a? 4 + . . . 



Here x Y is the given current in the first circuit, and <r 2 , x 3) . . . 

 are to be determined so as to make E a minimum. Now, E 

 being homogeneous in sc v w 2 , . . . , we have identically 



dE dE 



2E=^ lZ - +a? 8 t- + .... 



And since, when E is a minimum, 



dE c?E ,, . , 



jz> ^-,... all vanish, 



we see that 



2E(min.) =a;i g=(ll)^+(13)^ 9 . 



But if x 2 , x 3 , . . . had been all zero, 2E would have been equal 

 to (1 1) x 2 x . It is clear therefore that (1 2)x x x Q is negative ; 

 or, as (1 2) is taken positive, the sign of x 2 is the opposite of 

 that of x v 



