1909.] BETWEEN PARALLEL CONDUCTING CYLINDERS. 149 



be called the image of the polar line through OA. The point B, 

 thus defined, may be called the image polar point. The points A 

 and B, taken together, may be called the polar points of the diagram 

 with respect to the infinite plane and cylinder. 



Let P be any point in the plane of the diagram (Fig. 3). Then 

 let r' and r be the lengths of a pair of radii vectores BP, AP, drawn 

 from the polar points B, A, to P respectively. Let these distances 

 r'r be called the polar distances of the point P. Then the ratio m 

 of these polar distances will be : 



m = r'/r numeric (20) 



This ratio may be called the polar ratio, for purposes of reference. 

 The polar ratio will manifestly be a number greater than unity for 

 all points in the diagram above the infinite conducting plane Z'OZ. 

 It is a well known result that 



Ip 

 u = - log^ m abvolts (2 1 ) 



If a point be selected on the surface of the cylinder, having a poten- 

 tial Ml abvolts, and for convenience the lowest point of coordinates 

 ^1 and r = o, the polar distances of this point may be denoted by 

 r^' and r^ ; while their ratio may be denoted by w^ = r-l/'^x- Con- 

 sequently 



u^ = — log^ m^ abvolts (22) 



and eliminating /, p and 2.tt between (21) (22), we have 



11 = u. 



log^ in log 



log m iog,„;// 



1 —--- = ti, p^'^^— abvolts (23) 



The potential of the infinite plane is here reckoned as zero. It may 

 be observed that 



r^' a -\- d — <T a -\- d 

 in = — = = numeric ('24.) 



When the cylinder radius is very small, compared with the axial 

 distance d, d=a, and 



