67 



ad = W = cc = A* ; arid if a semidiameter OR of the one be cut 

 perpendicularly in P by a plane which touches the other, then will 

 OR be inversely as OP, so that OP x OR will be always equal to A:*. 



Let OR be a semidiameter of the ellipsoid whose semiaxes are 

 a', b', c ; and let «, /3, y, be the angles which it makes with them ; 

 then if x, y, z, be the coordinates of R, we have , 



-?1+ iC_+ -£--1 



o'2 ^ b'^ ~ c'^ "~ *' 



and therefore 



1 _ COS.* « 



, C0S.«/3 COS.«y 



—jX («• COS.* « + 6« COS.* /3 + C« COS.' v). 



But by the preceding lemma, since OP is perpendicular to the tan- 

 gent plane at Q, we have 



Hence 

 and therefore 



OP^ = a- COS." a ^ b^ COS.'' /3 -f- C° COS.' y. 



J 0F*_ 



OP X 0R = k'. 



3. If through the point of contact Q, the straight line OQ be 

 drawn to meet in N the tangent plane at R, it will meet it at right 

 angles. 



For the cosines of the angles made by OP with the semiaxes are 

 directly as the cosines of the angles made by OQ with them, and 

 inversely as the squares of a, b, c, (lem. 1. cor.) ; and the cosines of 

 the angles made with the semiaxes by a perpendicular to the tangent 

 plane at R, are directly as the cosines of the angles made with them 

 by OP, and inversely as the squares of a, b', c, or as the same cosines 

 and the squares of a, 6, c, directly ; that is, simply, as the cosines 



L 2 



