Electricity and Magnetism. 433 



of the circuits and of the moving parts of the systems in the 



two sets 



d BT 



d 



dt 



B^ = 

 871'" 



=0 



d 

 dt 



BT 

 B72'* 



=0 



d BT 



dt B .i'j 



BT 



B#i 



=0 



d BT 



^£ B<#2 



" B^2 



=0 



(4) 



(5) 



Here we suppose that only internal electromagnetic forces 

 BT/B^'i, BT/Bti'2, . • • , act on the system, and that there is no 

 ordinary dynamical force on the matter of the system. 

 There is no difficulty about including such forces as the 

 latter ; but the object at present is to obtain as simple as 

 possible a statement of the theory of a number of circuits 

 under their mutual action alone. Also it is assumed that 

 there are in the kinetic energy no terms involving products 

 of the form yx. See Maxwell, ' Electricity,' § 574. 



Multiplying equations (4) in order by 7^ y 29 ... , 7/, y 2 ' ... , 

 and the second set in order by 3c l9 i? 2 , ... , and adding, we 

 get for the interval of time dt, the equation 



dT=dTe+dT m =2(E 7 -n y *)dt. . . , (6) 



But the second set of equations (5) taken by itself gives 

 (since 



-/. d BT\ -», ■ ... 



.and the work dW done by the electromagnetic forces 

 BT/d#i, ...,isgivenby 



the relation 



dT m =dW (7) 



Hence we have 



dT e +d,W=S(Fjy-Ky-).dt. ... (8) 



