70 



6. An ellipsoid being cut by any plane through its centre, the 

 difference between the squares of the reciprocals of the semiaxes of 

 the section is proportional to the rectangle under the sines of the 

 angles which the plane of the section makes with the planes of the 

 two circular sections of the ellipsoid, — (See Fresnel's Memoir, 

 p. 150.) 



Let the plane which cuts the ellipsoid 

 intersect its circular sections in the lines 

 OR', OS', and let the principal section AOC 

 of the ellipsoid cut the circular sections in 

 OR and OS ; then OR, OS, OR', OS', will 

 be all equal to the mean semiaxis OB, and 

 hence the semiaxes OA', OC, of the section 

 A'OC will bisect the acute and obtuse angles 

 made by OR' and OS'. Let a plane through 

 OB and OA' intersect the principal plane AOC in the line OT. 

 Then by the nature of the ellipse, we have, in the ellipse A'OC, 



OR" 



Ol^ - \0C^ ~ OA^J ^'"•^ ^'^^'' 



and in the ellipse BOT, 



W - o^ = ( 0^ - O^) sin.' BOA'. 



Hence, observing that OB and OR' are equal, we have 



0C'»"~ OAi Vofii OT'J 



But the ellipse AOC gives 



1 \ sin.* BOA' 

 sin.» A'OR' 



[OR 



II - oTJ = ( ok - o!f) ^'^'"•^ ^^^ - '"'■' ^^^) = ( oZ?" - w) X 



sin. ROT sin. SOT. 



