263-267] Spherical Harmonics 227 



Thus the surface is an ellipsoid of which the centre is at the origin. It will 

 easily be found that r a + eTJ represents a spheroid of axis a-f-e, a ^, 



Zi 



and therefore of ellipticity ^-. 



265. We can treat these nearly spherical surfaces in the same way 

 in which spherical surfaces have been treated, neglecting the squares of the 

 small harmonics as they occur. 



266. As an example, suppose the surface r = a + S n to be a conductor, 

 raised to unit potential. We assume an external potential 



n + l 



where A and B have to be found from the condition that V= 1 when 

 r = a + S n . Neglecting squares of S n , this gives 



so that A = a, B = -, 



C!b 



a a n 



By applying Gauss' Theorem to a sphere of radius greater than a we 

 readily find that the total charge is a the coefficient of -. Thus the 



capacity of the conductor is different from that of the sphere only by terms 

 in S n 2 , but the surface distribution is different, for 



47TO- = y- = , if we neglect $ n 2 , 



the surface density becoming uniform, as it ought, when n = l, i.e. when the 

 conductor is still spherical. 



267. As a second example, let us examine the field inside a spherical 

 condenser when the two spheres are not quite concentric. Taking the centre 

 of the inner as origin, let the equations of the two spheres be 



r = b + eTJ. 



152 



