248 Methods for the Solution of Special Problems [CH. vm 



294. Equation (213) being of the second order, must have two inde- 

 pendent solutions. Denoting one by L, let the other be supposed to be Lu. 

 Then we must have 



= (A 



8a 2 



so that on multiplying the former equation by u, and subtracting from the 

 latter, 



,. d z u ~dLdu 

 L h 2 ^ = 0. 



9a 2 da da. 



Thus iwvfc ' dXj 



and the complete solution is seen to be 



where C and D are arbitrary constants. 



Accordingly, the complete solution of equation (211) can be written as 



This corresponds exactly to the general solution in rational integral 

 spherical harmonics, namely 



p n 



(C np "Pl (cos ff) + D np "Pl(cos 0)). 



Ellipsoid in uniform field of force. 



295. As an illustration of the use of confocal coordinates, let us examine 

 the field produced by placing an uninsulated ellipsoid in a uniform field of 

 force. 



The potential of the undisturbed field of force may be taken to be F = Fx, 

 or in confocal coordinates (cf. equation (201)) 



/(a* + \)(a* + ri(a*+v) 

 V" (6 2 -a 2 )(c 2 -a 2 ) 



