242 Methods for the Solution of Special Problems [OH. vm 



For instance, if the ellipsoid 



tf + & + tf = l 



is raised to unit potential, the potential at any external point will be given 

 by equation (206) provided we choose A and B so as to have F= 1 when 

 X = 0, and V = when X = oo . In this way we obtain 



F Jt 



r-cto, 



Jo A, 

 The surface density at any point on the ellipsoid is given by 



, 



47TCT = ^^ = - = /*! 



dn 8X dn d\ 



f x d\ 



Jo A A 



(208). 



Jo A A 



Thus the surface density at different points of the ellipsoid is proportional 

 to h lf 



284. The quantity h-^ admits of a simple geometrical interpretation. 

 Let I, m, n be the direction-cosines of the tangent plane to the ellipsoid at 



FIG. 79. 



any point X, ^t, v, and let p be the perpendicular from the origin on to this 

 tangent plane. Then from the geometry of the ellipsoid we have 



x)?72, 2 + (c 2 + X)n 2 ............ (209). 



Moving along the normal, we shall come to the point X + d\, p, v. The 

 tangent plane at this point has the same direction-cosines I, m, n as before, 



but the perpendicular from the origin will be p + dp, where dp = -j- . To 



"i 



