216 THEORY OF NEWTONIAN FORCES. [FT. I. CH. V. 



At infinity a- = oo , and V and its derivatives accordingly 

 vanish. 



Hence the value of V found satisfies all the conditions. 



115. Ivory's Theorem. If x, y, z is a point on the 

 ellipsoid 



(\\ \y \ z i 



VV o ' I. o ' o ^j 



j.1 . Q"> t>2 Co 



the point as , jf jn, - 



Cli PI Ci 



lies on the ellipsoid 



These will be called corresponding points. We shall now 

 assume that these two ellipsoids are COD focal, and (2) the smaller. 

 Then 



The action of (2) on the external point x, y, z is 



I"* 5 du 



X, = - &ra,ta0 J ^ (aj , + B) ^-^-j-^, + 



where 



a; 2 2 ^ 

 and since _ + |L + _ = 1 , 



we must have a = X. 



If now we substitute 



a 2 2 = w' + cr, 



/.'T^ 



Now the attraction of the ellipsoid (i) on the interior point 



a 2 &a c 2 . 

 a? - , y-r, z- is 



b fc c i 



TT o L a 



A j = ZTraACi a? 



(a? + u) v(a! 2 + u) (b-? + u] (d 2 + w) 



