344 ELECTROKINETICS. [PT. II. CH. VIII. 



This formula is important in the case of standards of resistance 

 formed of tubes filled with mercury, the varying diameter of the 

 tube being determined by a calibration. If the conductor is 

 homogeneous, X is constant, and if the cross-section is constant, 



Xw 



or the resistance of a uniform wire is proportional to its length 

 and inversely to its cross-section. 



175. Non-linear Homogeneous Conductors. In the case 

 of homogeneous conductors, X being constant, the equation of flow, 

 5 162 (5), becomes 



a 2 F 9 2 F 



or the potential is harmonic. Consequently every theorem on 

 harmonic functions applies to the potential in this case, and 

 every method of solving problems of electrostatic distribution may 

 be applied to the solution of problems of steady flow. We must 

 have the electrodes of the conductor given. Now by the equation 

 of Ohm's Law it is evident that the effect of increasing the 

 conductivity of any portion of a conductor is to make the potential 

 vary less rapidly there, the current being given. If then a portion 

 of the conductor be made infinitely conducting its potential will 

 become constant throughout. Accordingly if we introduce a thin 

 plate of infinitely conducting material, this will form an equipo- 

 tential surface and may be taken as an electrode for the conductor. 

 This supposition will be made in the following examples. Since in 

 the electrostatic problem the capacity is given by 



T7- J 



" F.-F, 



and in the problem of flow the conductance by 



F 2 - F x ~ R ' 



we find that the conductance of a portion of a homogeneous con- 

 ductor between two electrodes is equal to 4?rX times the capacity 

 of a condenser whose plates have the geometrical form of the 

 electrodes of the conductor, and whose dielectric occupies the 



