1909.] BETWEEN PARALLEL CONDUCTING CYLINDERS. 147 



P 



h = ^. 



coth-M ^j abvolts (14) 



u 



where / and tt have the same meanings as above, and the potential 

 of the plane Z'OZ is reckoned as zero. 



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

 OY , distant y^ cm. from O, and above the polar point A, is: 



^ = — coth- \-\ abvolts (15) 



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

 W3, and 3/3 be the distance of the highest point of the cylinder from 

 the plane, the potential at any other point on the median line, above 

 the cylinder, and distant 3'^ cm. from the plane, will be : 



coth~^ iyjci) 



^^^ = ^^ — Ti — VT T-\ abvolts (16) 



^ 3 coth-^ {y^la) ^ ' 



Potentials at Points Outside the Cylinder and off the Median 

 Line. — If the point in the plane Z'YZ at which the potential is 

 required, lies ofif the median line OY , the potential may be expressed 

 either : 



(0) In terms of rectangular coordinates z and 3' of the point. 



{b) In terms of the ratio of radii vectores to the point, from the 

 polar point A, and from its image. 



(a) Potential in Terms of Rectangular Coordinates. — Let P, 

 Fig. 3, be the point whose potential is required, and whose rectan- 

 gular coordinates are y and z, measured respectively along the me- 

 dian line OY , and the line OZ in the infinite conducting plane. 

 Then u, the potential of F, is : 



IP 



u = — 

 27r 



(2ay \ 

 -s— — 2— — s I abvolts (17) 



« +y + .^ / 



where /, p and a have the values previously assigned", and the poten- 

 tial of the plane Z'OZ is reckoned as zero. Eliminating Ip/i: with 

 the aid of (11), we have: 



tanh-' 





u = u. 7 — r— pT — 1-^ abvolts (18) 



^ 2 tanh-^ O'l/^) 



