279-283] Confocal Coordinates 241 



and clearly h 2 and h s can be obtained by cyclic interchange of the letters 

 X, /ji and v. 



282. If for brevity we write 



A A = V(a 2 + X) (6 2 + X) (c 2 + X), 

 we find that 



so that by substitution in equation (198), Laplace's equation in the present 

 coordinates is seen to be 



8 \t ^ A * 8F 1 _!_ 8 \( ^ A * 8F lo_ a k N A - aF l 



9^r-^AA^} + 97r- x) A^^t + a^{( x -^A^ M ^} =o 



............... (203). 



On multiplying throughout by A^A^A,,, this equation becomes 



O*-')^*^)^^:.^--^!^^ 



............... (204). 



Let us now introduce new variables or, /3, 7, given by 



A ' 



*-* 



r) f\ 



then we have = A x 5- ; 



dot. dX 



and equation (204) becomes 



Distribution of Electricity on a freely-charged Ellipsoid. 



283. Before discussing the general solution of Laplace's equation, it will 

 be advantageous to examine a few special problems. 



In the first place, it is clear that a particular solution of equation (205) is 



V=A + BOL (206), 



where A, B are arbitrary constants. The equipotentials are the surfaces 

 a = constant, and are therefore confocal ellipsoids. Thus we can, from this 

 solution, obtain the field when an ellipsoidal conductor is freely electrified, 

 j. 16 



