348 ELECTROKINETICS. [FT. II. CH. VIII. 



is constant by hypothesis, we have 



dV/dz = const., 



and if dV/dz is to have the same value on the upper side, V must 

 there be the same as the potential due to a fictitious (non-equi- 

 potential) distribution on a disk of radius a of constant density 



_1_ 9F_ JL^ q 



~ ZTT dz~ 2?r X ' 



The mass of such a distribution would be 



a 2 q 



m = jra 2 cr = - . 

 &\ 



The resistance of the upper side may be calculated by Joule's Law, 



TT o~ -5- r~ 



H JJJ( \dxj \dyj \c)z 



The integral in the numerator being through one-half of infinite 

 space is 8?rX times one-half the energy of the distribution on the 

 disk. The integral in the denominator is 4?rX times one-half the 

 mass of the disk. Consequently 



W 



where W is the whole energy of the distribution of the disk. This 

 energy is very easily calculated. The potential at the edge of a 

 disk of radius p with constant surface density a- is 



r ffdS 



HjT- 



If we introduce polar coordinates, the origin being the attracted 

 point on the edge, and being the angle included between r, the 

 radius to the point of integration and the diameter through the 

 origin, this becomes 



r-2 rr 

 = <r / 



J *Jr= 



