246 



Methods for the Solution of Special Problems [CH. vm 



Plane disc 



Curved disc arc 10 



Curved disc arc 20 



Bowl arc 270 C 



Bowl arc 340 



Discussing these results, Lord Kelvin says : " It is remarkable how slight an amount 

 of curvature produces a very sensible excess of density on the convex side in the first two 

 cases (10 and 20), yet how nearly the mean of the densities on the convex and concave 

 sides at any point agrees with that at the corresponding point on a plane disc shewn in 

 the first column. The results for bowls of 270 and 340 illustrate the tendency of the 

 whole charge to the convex surface, as the case of a thin spherical conducting surface with 

 an infinitely small aperture is approached." 



ELLIPSOIDAL HARMONICS. 

 292. We now return to the general equations (205), namely 



(211), 



and examine the nature of the general solutions of this equation. 

 Let us assume a tentative solution 



V=LMN, 



in which L is a function of X only, M a function of //, only, and N a function 

 of v only. Substituting this solution the equation reduces to 



, 1 d 2 L , 1 d*M ^ 1 





