244 Methods for the Solution of Special Problems [OH. vm 



Elliptic Disc. 



287. In the preceding analysis, let a become vanishingly small, then 

 the conductor becomes an elliptic disc of semi-axes b and c. 



The perpendicular from the origin on to the tangent-plane is given, as in 

 the ellipsoid, by 



1 



a 4 6 4 c 4 



and when a is made very small in the limit, this becomes 



1 a 2 



D = = 



" 1,2 v ' 



a 4 2 c 2 



so that the surface density at any point #, y in the disc is proportional to 



Circular Disc. 



288. On further simplifying by putting b = c, we arrive at the case of a 

 circular disc. The density of electrification is seen at once from expression 

 (210) to be proportional to 



and therefore varies inversely as the shortest chord which can be drawn 

 through the point. 



Moreover, when a = and b = c, we have A A = (c 2 + X) Vx, so that 



r d\ 2 ^ / c \ , rd\ TT 



- = - tan- 1 -= arid = - . 



J A A A c WX/ Jo A A c 



2c 

 Thus the capacity of a circular disc is , and when the disc is raised to 



potential unity, the potential at any external point is 



2 tan- f-fV 



7T V\/X/ 



where X is the positive root of 



a* f+z* 

 X c 2 + X 



