of Force between Electric Currents. 465 



where p is density, y potential of mass*. This equation occurs 

 in the electromagnetic theory of light, giving one of the two 

 expressions for u. from which arises the equation 

 Jr ct 2 F d?F 



2o. Again, it is a known property of the potential of mass, 

 that if the potential of mass A has the same value as that 

 of mass B at every point of a surface completely enclosing 

 both, it has the same value at all points in space outside that 

 surface. 



In like manner we conclude that if F, the vector potential 

 of the x component of a system of currents, has the same value 

 as F' (that of another system) at all points of a surface com- 

 pletely enclosing both systems, then F has the same value as 

 F' at all points beyond that surface. Hence can be easily de- 

 duced, by way of illustration, the theory of magnetic images 

 as follows. 



Let there be an infinite conducting plane, and let a deter- 

 minate system of electric currents be suddenly generated 

 parallel to and wholly on one side of the plane. Let u, v be 

 the two components of current at any point in the system, 

 F, G the components of vector potential of the system. By 

 the principle of least kinetic energy a system of currents will 

 be excited or "induced" in the plane, such as to make the 

 whole energy the least possible consistently with the given 

 currents in the given system. Let u f , v' be the components of 

 these currents at any point in the plane ; F', G' the corre- 

 sponding components of vector potential. Then for the whole 

 energy we have 



2T=/* JJ { { F + F)« + (G + G 7 > } do over the system, 



+ fj,]]{(F + W)v/ + (G + G/y } da over the plane. 

 And in order for T to be a minimum, given u and v at all 



dT dT 



points in the system, we must have -=- - f = 0, -=— f = at every 



point in the plane. That is, F=— F', G= — G' at every 

 point in the plane ; and this being true at every point in the 

 plane, must be true at all points in space beyond the plane. 



The system of currents induced in the plane thus at all 

 points beyond the plane exactly neutralizes the given system. 

 It has, in fact, the same effect as the given system reversed in 

 direction. 



* The constant fx depends on the nature of the medium where the cur- 

 rents are situated, and stands in the same relation to the current as -v 



(where k is the specific inductive capacity of the dielectric) stands to free 

 electricity. 



