326 ELECTROKINETICS. [PT. II. CH. VIII. 



and as this must be true for any portion of space fulfilling the 

 above conditions, we must have everywhere in such regions 

 du dv dw . 



(2) r- + 5- + 3- = 0. 



dx dy dz 



This is called the equation of continuity, and shows that the 

 current density is a solenoidal vector. Current lines and tubes 

 accordingly possess properties similar to those of tubes of force in 

 the case of equilibrium. 



The law of Ohm is identical with that stated by Fourier* for 

 the conduction of heat, and connects the current density with the 

 potential, the corresponding quantity for heat being the tempera- 

 ture. If the conductor be isotropic, that is if its properties are at 

 each point the same for all directions, the direction of the current 

 is the same as that of the electrostatic field, and their magnitudes 

 are proportional, the factor of proportionality depending on the 

 physical properties of the conductor at each point. If n is the 

 normal to an equipotential surface at the point in question drawn 

 in the direction of the current, we have 



(3) ff -vr~x ; 



as the mathematical statement of Ohm's Law. The factor of pro- 

 portionality X is called the conductivity, and its reciprocal the 

 specific resistance, or resistivity of the conductor. If \ is the same 

 at all points of the conductor, the conductor is said to be homo- 

 geneous, if X is variable, the conductor is heterogeneous. 



The above equation is equivalent to the three 



3F SF 3F 



(4) u= -\^~ , v X ^ , w = \^-. 



dx dy dz 



Inserting these in the equation of continuity, 



(5) fa V /v fa) " dy V' v dy 



If the conductor is homogeneous, this becomes A F= 0. Hence 

 the density p is zero, or there is no free electricity in any portion of 

 a homogeneous conductor in the state of steady flowf. 



* Fourier, Theorie analytique de la chaleur, 1822. 



t Kirchhoff. "Ueber eine Ableitung der Ohm'schen Gesetze, welche sich an die 

 Theorie der Elektrostatik anschliesst." Pogg. Ann., Bd. 78, 1849. Ges. Abh., p. 49. 



