141, 142] ELLIPSOID. CONCENTRIC SPHERES. 275 



If we now make z 0, we get the density on an elliptical disc 



e 



and if a b, 



4-rra Va 2 - r 2 ' 

 for a circular disc of radius a. At the edge of the disc, 



and the density is infinite, so that this case is not physically 

 possible. It is however of considerable theoretical importance. 



For the case of the circular disc the potential at any point 

 becomes 



Tr e f ds e (TT . Vx) e , a 



V = ^ ._ = - \- - tan" 1 \ = - tan" 1 -= , 



2 J \ (a 2 + s) v s & (2 a j a Vx 



where X is the greatest root of the quadratic 



^ "*" ^ 2 4. z - i 

 a 2 + X X 



142. Concentric Spheres. Suppose we have a sphere of 

 radius J^ x , surrounded by a concentric spherical shell of radii R% 

 and R s . In the space between the conductors and outside of the 

 outer, V satisfies the equation, 88 (7), 



whose integral is 



dVA 



. 

 r 



If Fj is the potential of the inner sphere, F 2 of the shell, 



182 



