SUPERPOSITION OF PERMANENT STATES. 189 



the two states, .the potentials at each point add together, and the 

 perpendicular force upon the element dS becomes 



- + 



By hypothesis, the flow across the element dS has become 



AdS +-A'dS. 



If we suppose the initial conditions to be identical, the flow becomes 

 2A^S, and the force is thus doubled ; the flow of electricity is there- 

 fore proportional to the flow of force. 



If the state is permanent that is, if the potential is unchanged 

 at each point the flow across any surface-element is constant. If 

 the state is variable, the potentials vary with the time, and the flow 

 d 2 Q across an element of surface during the time dt is 



203. IN THE PERMANENT STATE THE DENSITY IN THE IN- 

 TERIOR OF A CONDUCTOR is NULL. The flow of electricity cannot 

 accumulate in the interior of a closed surface, without modifying the 

 potential, just as an increase in the flow of heat would produce there 

 a rise of temperature. 



If the permanent state has been attained for a system of con- 

 ductors, the total flow which each element receives is zero. If the 

 algebraical sum of the flows which enter a volume-element dxdydz is 

 equal to zero, we find again Laplace's equation : 



p being the cubical density of electricity at the point in question, 

 we have always 



AV = - 47173, 

 from which 



/> = (). 



The law of the proportionality of the flow of electricity to the 

 flow of force leads directly to the same conclusion. For, if the state 



