of Surfaces of the Second Degree, 



191 



8. There are four points of 

 an ellipsoid (the umbilics 

 namely), which are the vertices 

 of cones of revolution standing 

 on one of its focal conies. 



9. The locus of such points 

 in a system of coaxal and con- 

 focal ellipsoids is the conjugate 

 focal conic, which is a hyper- 

 bola. 



10. The focal hyperbola is 

 also the locus of the vertices of 

 cones of revolution enveloping 

 a given ellipsoid. 



11. The intersection of three 

 rectangular planes, tangent re- 

 spectively to three coaxal and 

 confocal ellipsoids, moves on a 

 sphere concentric with the 

 ellipsoids. 



12. The difference of the 

 squares of perpendiculars let 

 fall from the common centre 

 on parallel tangent planes to 

 two coaxal and confocal ellip- 

 soids is constant. 



13. If through a line lying 

 in the plane of one of the focal 

 conies of a system of coaxal 

 and confocal ellipsoids, a tan- 

 gent plane be applied to each 

 of them, the locus of the points 

 of contact is a circle. 



When the ellipsoid to whic 



8. There are four tangent 

 planes to an ellipsoid intersect- 

 ing one of its cyclic cylinders 

 in curves which lie on cones 

 of revolution, having their 

 common vertex at the centre 

 of the surface. 



9. The series of such tan- 

 gent planes to a system of co- 

 axal and biconcyclic ellipsoids 

 envelope the conjugate cyclic 

 cylinder, whose base is a hy- 

 perbola. 



1 0. The same cylinder is the 

 envelope of a series of planes 

 cutting out of an ellipsoid sec- 

 tions which lie on cones of re- 

 volution, whose common ver- 

 tex is the centre of the system. 



11. If three radii, which are 

 two by two at right angles, be 

 drawn from the common cen- 

 tre of three coaxal and bicon- 

 cyclic ellipsoids, and through 

 the points where these radii 

 respectively meet the threesur- 

 faces, a plane be made to pass, 

 this plane will envelope a 

 sphere concentric with the el- 

 lipsoids. 



12. The diiference of the 

 squares of the reciprocals of 

 coincident radii in two coaxal 

 and biconcyclic ellipsoids is 

 constant. 



13. If a line be parallel to 

 the axis of one of the cyclic 

 cylinders of a system of coaxal 

 and biconcyclic ellipsoids, and 

 at the points where it meets 

 each of these surfaces, tangent 

 planes be applied to them, 

 these planes will envelope a 

 cone of the second degree. 

 1 a cyclic cylinder belongs be- 



