146 KENNELLY— THE LINEAR RESISTANCE [April 24, 



Potential Distribution. 



On the Median Line Beneath the Cylinder. — It is well known 

 that the flow of electric current, and the distribution of potential, 

 between the conducting cylinder and the plane, are such as might be 

 produced by removing the conducting cylinder and substituting a 

 conducting polar line at A, parallel to the plane. The point A lies 

 on the line OC, and at a distance a from the plane defined by the 

 relation 



a^=(T sinh Y =\/d- — o--. cm. ( 10) 



The values of the polar ratio a/a- are given in the table in column 

 VII. for each of the selected ratios d/a- up to J/(r^^50, beyond 

 which the difference between a/a- and d/a- is less than i part in 

 5,000. For most practical purposes, it is, therefore, sufficient to 

 regard, the polar line as coinciding with the cylinder axis when the 

 distance of that axis from the plane exceeds 50 radii. 



In the steady state of flow, the potential at any point _Vi on the 

 line OA (Fig. 3) distant 3'^ cm. from 0, will be 



?/j = /^tanh-i( -^ j - abvolts (11) 



where / is the current strength per linear cm. of the system in 

 absamperes, the potential of the plane Z'OZ being taken as numer- 

 ically zero. 



Similarly, the potential at any other point 3^ on the median line 

 OY, below A, distant 3^ cm. from 0, will be: 



//, = / tanh-^ ( ' ^ ) abvolts (12) 



Consequently, if the potential of the surface of the cylinder be u^, 

 and y^ be the distance of the lowest point of the cylinder from the 

 plane, the potential of any other point on the line OA between the 

 cylinder and the plane, distant 3-0 cm. from the latter, will be : 



tanh-* (Jz/^) u 1^ / X 



«9= ?^ I — , 1 ) , ; abvolts (13) 



2 I tanh-' (ji/«) ^ ^ 



Potentials on the Median Line Above the Cylinder. — In the 

 steady state of flow, the potential at any point 3*3 on the median line 

 OY, and distant 3', cm. from O, above the polar point A, is: 



