L 



BY A STATIONARY ELECTRIC CURRENT. 203 



The above-mentioned contradiction therefore disappears ; for 

 according to this signification of the function V, the equation 

 V = const., which in (1) denotes that no current exists, is the 

 same as that which in electro-statics is known as the con- 

 ditional equation for a state of equilibrium. 



Further, this signification of V being adopted, it is easy to 

 determine, as KirchhofF has shown, where the free electricity is 

 distributed in the conductor during a stationary current. For, 

 in order that a current may be stationary, the quantity of elec- 

 tricity contained in each element of space must be constant, 

 hence the quantities of electricity entering and issuing from any 

 element must be equal. Let dxdydz be such an element, 

 situated at the point {x, y, z), then, according to equation (1), 

 the quantity of electricity entering the element through the 

 first of the two surfaces dy dz, in the unit of time 



= k.dydz-^, 



and the quantity issuing from the element through the opposite 

 surface 



hence the excess of the latter over the former 



= kdxdy dz -j-^. 

 ax~ 



Similarly, for the pair of surfaces dx dz we obtain the excess 



d^Y 



df 

 and for the pair of surfaces dx dy, 



k dx dy dz -^-^* 



The sum of these three last expressions gives the excess of the 

 total quantity of electricity issuing from the element over the 

 quantity entering it in the unit of time, and as this excess must 

 be null, we have 



d'^Y d^N d^Y 

 dx^^dy^ ^ dz' 

 From a well-known theorem of the potential function, how- 

 ever, it follows from this equation, that the point {x, y, z) must 

 be situated without the mass, of which V is the potential func- 



Q2 



k dxdydz g. 





