Medium, according to the Principles of FresneL 3 



ellipsoids coincide in direction, and be reciprocally proportional, 

 so that aa' = W = cc' = k z ; and 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 k*. 



Let OR be a semidiameter of the ellipsoid whose semiaxes 

 are of, b', c' ; and let a, /3, 7, be the angles which it makes with 

 them ; then if x, y, z, be the co-ordinates of R, we have 



and therefore 



1 _ cos 2 a cos 2 /3 cos 2 7 



Qff ~ a '2 fr2 + c "i 



Fig. 2. 

 = T- 4 (a 2 cos 2 a + b 2 cos 2 )3 + c 2 cos 2 7.) 



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

 tangent plane at Q, we have 



OP 2 = a* cos 2 a + b 2 cos 2 /3 + c 2 cos 2 7. 

 Hence 



1 OP 2 



and therefore 



OPx OR = k\ 



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

 drawn to meet in JV the tangent plane at P, it will meet it at 

 right angles. 



For the cosines of the angles made by OP with the semi- 

 axes 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 perpen- 

 dicular 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", >', c', or as the same cosines and the squares of a, b, c, di- 

 rectly ; that is, simply, as the cosines of the angles made by OQ 



