150 KENNELLY— THE LINEAR RESISTANCE [April 24, 



r ' 2d D 

 VI. = - == — = — numeric (25) 



^ r^ a (T ^ ' 



It follows from the preceding equations that the equipotential 

 surfaces in an infinite plane-cyHnder system are all cylinders having 

 their axes situated on the median line. If u-^ be the potential of the 

 conducting cylinder, and if we denote by F^ the value of the distance 

 factor Y for this cylinder, according to formula (3), or to column 

 II. of the table, then the distance factor Y of any cylindrical equi- 

 potential surface whose potential is u becomes 



71 



F= Fj— numeric (26) 



We have for any such cylinder the equations of condition : 



F=cosh-Hc?A) =sinh-i(a/cr) =tanh-i(a/d) =cotli-i((//a) 



= 2 tanh"^(3;/a) numeric (27) 



whence d, the axial distance, or y coordinate, of the cylinder whose 

 potential is u, will be along the median line OY : 



d= — f—,r\ ''"'• (^7) 



tanh(F,-) 



and the radius a of this equipotential cylinder is : 



a 



sinh 



(-■9 



cm. (28) 



The coordinate 3; of the lowest point of any such equipotential 

 cylinder will be: 



/ F \ / F. u \ 



= a tanh ( — j = ^ tanh ( — I cm. (30) 



so that 



tanh" 



V W + I / 



^^ = ^^— T^tT^T — 1-V- abvolts (31) 



tanh \)\\(-i) 



