153, 154J ELLIPSOIDAL EQUIPOTENTIALS. 413 



From the geometrical definition of A, 



lim A = 1. 



Now consider, for simplicity, a point on the X-axis, where 



$1 = x = r. The denominator becomes infinite in A 2 , that is, r 5 , and 

 so does the numerator. Hence 



so that 



154. Chasles's Theorem. We have now found the potential 

 due to a mass M. of such nature that its equipotential surfaces are 

 confocal ellipsoids, but it remains to determine the nature of the 

 mass. This may be varied in an infinite number of ways; we will 

 attempt to find an equipotential surface layer. By Green's theorem, 

 129, 11), this will have the same mass as that of a body within 

 it which would have the same potential outside. 



If we find the required layer on an equipotential surface S, since 

 the potential is constant on S, it must be constant at all points 

 within, or the layer does not affect internal bodies. 



The surface density must be given by 10), 129, 



(3 = --> where m is the outward normal to A, 



and 



3V _ dV dl __, dV 



d^i ~ 111 fh^ ~ l ~dl' 

 Now since 



15) (? = - di 



%TC d'k 

 Since V is a function of A alone, the same is true of -=T-> which for 



a constant value of A is constant. Hence tf varies on the ellipsoid S 

 as d%. Therefore if we distribute on the given ellipsoid S a surface 

 layer with surface density proportional at every point to the perpen- 

 dicular from the origin on the tangent plane at the point, this layer 

 is equipotential, "and all its equipotential surfaces are ellipsoids confocal 

 with it. Consequently if we distribute on any one of a set of 

 confocal ellipsoids a layer of given mass whose surface density is 

 proportional to d the attraction of such various layers at given 



