508 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. 



Differentiating these equations respectively by x, y, z and 

 adding we obtain 



so that the total current is solenoidal, like the flow of an incom- 

 pressible fluid. Integrating (15) through a portion of space r 

 bounded by a closed surface S, 



= [u cos (nx) + v cos (ny) + w cos (nz)} dS, 

 which by 182 (17) becomes 

 (17) gj J J J pdr = g- = II (M cos (nx) + v cos (ny) + w cos (ras)} d 



That is, the increase of charge of any portion of space is equal 

 to the electricity brought in by conduction. This agrees perfectly 

 with our previous conceptions. Our statement made in 129 that 

 electricity is not incompressible is also reconcilable with Maxwell's 

 statement that the total current, the resultant of the conduction 

 and displacement currents, is like the flow of an incompressible 

 fluid. 



By the analogy between the equations (5) and (12) we 



1 r)9^ 



might call the vector j -^r * ne magnetic displacement current. 



Magnetic conduction-currents do not exist, although they have 

 been introduced into the equations by Heaviside* for the sake 

 of symmetry. 



245. Complete System of Equations for Media at rest. 



We may now collect all the fundamental equations of the theory 

 as it has been developed. Before doing this it will be convenient 

 to make a slight change in our units. It will be recalled that in 

 the whole of Part III since the introduction of the electro- 

 magnetic system of units we have considered all quantities, 

 whether electrical or magnetic, to be measured in that system. 

 Up to the present this has been most convenient, and in prac- 

 tical cases dealing with electro-magnetism and electro-magnetic 



* " Electromagnetic Induction and its Propagation." Electrical Papers, Vol. i. 

 p. 441. 



