283-286] Confocal Coordinates 243 



obtain dp we differentiate equation (209), allowing \ alone to vary, and so 

 have 



2pdp = d\ (I 2 + ra 2 + ?z 2 ) = d\. 



Comparing this with dp = -j- , we see that h = 2p. 



*h 



Thus the surface density at any point is proportional to the perpendicular 

 from the centre on to the tangent plane at the point. 



In fig. 79, the thickness of the shading at any point is proportional to 

 the perpendicular from the centre on to the tangent plane, so that the 

 shading represents the distribution of electricity on a freely electrified 

 ellipsoid. 



It will be easily verified that the outer boundary of this shading must be 

 an ellipsoid, similar to and concentric with the original ellipsoid. 



285. Replacing /^ by 2p in equation (208), we find for the total charge E 

 on the ellipsoid, 



Jo 



Since l\pdS is three times the volume of the ellipsoid, and therefore 

 equal to 4>7rabc, this reduces to 



r= 



Jo A A 



Since the ellipsoid is supposed to be raised to unit potential, this quantity 

 E gives the capacity of an ellipsoidal conductor electrified in free space. 



286. The capacity can however be obtained more readily by examining 

 the form of the potential at infinity. At points which are at a distance r 

 from the centre of the ellipsoid so great that a, b, c may be neglected in 

 comparison with r, X becomes equal to r 2 , so that A* = r%, and 



I 



r 

 Thus at infinity the limiting form assumed by equation (207) is 



2 



and since the value of V at infinity must be , the value of E follows at 

 once. 



162 



