208 THEORY OF NEWTONIAN FORCES. [PT. I. CH. V. 



From the geometrical definition of X, 



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

 S A =#=r. The denominator becomes infinite in X^, that is, r 5 , and 

 so does the numerator. Hence 



lim 



so that 

 (12) - -' & 



110. 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 Chasles's 

 theorem, 84 (n), this will have the same mass as that of a 

 body within it which would have the same potentials outside. 



If we find the required layer on an equipotential surface 8 t 

 since the potential is constant on 8, it must be constant at all 

 points within, or the layer does not affect internal bodies. 



The surface density must be given by 84 (10), 



1 8F 



" ~~ ~r~ o > where n^ is the outward normal to X, 

 4?r dn x 



and 



a_F == ^F_ax ==/ , dv 



Now since h^ 28x, 



..a'**" 



Since F is a function of X alone, the same is true of -= , which for 



a constant value of X is constant. Hence cr varies on the ellipsoid 

 8 as S\. Therefore if we distribute on the given ellipsoid 8 a 

 surface layer with surface density proportional at every point to 

 the perpendicular from the origin on the tangent plane at the 

 point, this layer is equipotential, and all its equipotential surfaces 



