294, 295] Ellipsoidal Harmonics 249 



This is of the form 



V= CLMN 



where C is the constant F(b*-a 2 ) ~^(c 2 -a 2 )~^, and L, M, N are functions of 

 X only, /JL only and v only, respectively, namely L = Va 2 + X, etc. 



Since F= ZIW is a solution of Laplace's equation, there must, as in 294, 

 be a second solution V Lu . MN, where 





The upper limit of integration is arbitrary : if we take it to be infinite, 

 both u and Lu will vanish at infinity, while M and N are in any case finite 

 at infinity. Thus Lu . MN is a potential which vanishes at infinity and is 

 proportional (since u is a function of X only) at every point of any one of the 

 surfaces X = cons., to the potential of the original field. Thus the solution 



V CLMN + DLu . MN (215) 



can be made to give zero potential over any one of the surfaces X = cons., by 

 a suitable choice of the constant D. 



For instance if the conductor is X = 0, we have, on the conductor, 



f 00 d\ 



* Jo <y 



Thus on the conductor we have 



o \u> 



The condition for this to vanish gives the value of D, and on substituting 

 this value of D, equation (215) becomes 



u 



(216). 



This gives the field when the original field is parallel to the major axis 

 of the ellipsoid. If the original field is in any other direction we can resolve 

 it into three fields parallel to the three axes of the ellipsoid, and the final 

 field is then found by the superposition of three fields of the type of that 

 given by equation (216). 



