2 2 Geometrical Propositions applied 



eoids having a common centre at the point 0, and their semi- 

 axes coincident in direction, and connected by the relation 

 aa' = bb f = cc = k z ; where a, J, c, are semiaxes of one ellipsoid in 

 the order of their magnitude, a being the greatest ; and a', b', c', 

 those of the other ellipsoid, a being the least. The mean semi- 

 axes b and b' coincide, and the circular sections of both ellipsoids 

 pass through the common direction of b and b'. 



5. It has also been shown with regard to those ellipsoids, 

 that if Q and R be reciprocal points on the surfaces of aba and 

 a'b'c' respectively, and if a right line Oqr, perpendicular to the 

 plane QOR, cut the first ellipsoid in q and the second in r, the 

 lines OQ and Oq will be the semiaxes of the section made in the 

 ellipsoid abc by a plane passing through them ; and the lines 

 OR and Or, in like manner, will be the semiaxes of the section 

 made in the other ellipsoid d V c' by the plane in which they lie. 



6. It may further be remarked, that if the radius OQ in one 

 of the reciprocal ellipsoids describe a plane, the corresponding 

 radius OR will describe another plane. For the planes touching 

 the ellipsoid abc in the points Q will all be parallel to a certain 

 right line, and therefore the perpendiculars OR to these tangent 

 planes will all lie in a plane perpendicular to that right line. 

 These two planes, containing the reciprocal radii, may, for 

 brevity, be called reciprocal planes. 



When two reciprocal radii lie in a principal plane, at right 

 angles to a semiaxis of the ellipsoids, it is evident that two planes 

 intersecting in this semiaxis, and passing through the reciprocal 

 radii, are reciprocal planes. 



7. THEOREM II. If three right lines at right angles to each 

 other pass through a fixed point 0, so that two of them are con- 

 fined to given planes, the plane of these two, in all its positions, 

 touches the surface of a cone whose sections parallel to the given 

 planes are parabolas ; while the third right line describes ano- 

 ther cone, whose sections parallel to the given planes are circles. 



Let the plane of the figure (Fig. 10), supposed parallel to one 

 of the given planes, be intersected by the other given plane in 

 the right line MN ; and let OQ be perpendicular to the latter 



