A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 567 



If there are two magnetic poles m^ and m. 2 producing potentials ^ and < 2 

 in the field, then if m 3 is moved a distance dx, and is urged in that direction 

 by a force X, then the work done is Xdx, and the decrease of energy in the 

 field is 



and these must be equal by the principle of Conservation of Energy. 



Since the distribution ^ is determined by m l} and <, by m a , the quantities 

 and ^ a m 2 will remain constant. 



It can be shewn also, as Green has proved (Essay, p. 10), that 



so that we get 

 or 



- 



where c^ represents the magnetic intensity due to m r 



(K). 



Similarly, Y= mJ3 lt 



Z = m 3 y 1 . 



So that a magnetic pole is urged in the direction of the lines of magnetic 

 force with a force equal to the product of the strength of the pole and the 

 magnetic intensity. 



(78) If a single magnetic pole, that is, one pole of a veiy long magnet, 

 be placed in the field, the only solution of <f> is 



*=-~f ( 41 )> 



where m l is the strength of the pole, and r the distance from it. 

 The repulsion between two poles of strength in\ and m t is 



(42). 



In air or any medium in which \L = 1 this is simply -^ , but in other 



media the force acting between two given magnetic poles is inversely propor- 

 tional to the coefficient of magnetic induction for the medium. This may be 

 explained by the magnetization of the medium induced by the action of the 

 poles. 



