190 Mr. J. K. Ingram on New Properlies 



one standing on a focal conic 

 of a surface of the second de- 

 gree, and the other enveloping 

 that surface, these two cones 

 will have the same principal 

 axes and the same focal lines. 



4. If any point be made the 

 common vertex of two cones 

 standing on the two focal co- 

 nies of a surface of the second 

 degree, these two cones will 

 have the same principal axes 

 and the same focal lines. 



5. If any point be made the 

 vertex of a cone circumscri- 

 bing an ellipsoid, the tangent 

 plane applied at the point to a 

 coaxal and confocal ellipsoid 

 described through it, is a prin- 

 cipal plane of that cone. 



6. And the tangent planes 

 applied at the same point to 

 the other two coaxal and bi- 

 concyclic surfaces of the second 

 degree described through it, 

 are the other two principal 

 planes of that cone. 



7. If any point on an ellip- 

 soid be made the vertex of a 

 cone standing on one of its 

 focal conies, the tangent plane 

 to the surface at the point will 

 be a principal plane of that 

 cone. 



degree and one of its cyclic cy- 

 linders in two curves, and the 

 centre of the surface be made 

 the common vertex of two 

 cones, passing respectively 

 through those two cui'ves, the 

 two cones will have the same 

 principal axes and the same 

 cyclic planes. 



4. If a plane be drawn cut- 

 ting the cyclic cylinders of a 

 surface of the second degree in 

 two curves, and the centre of 

 the surface be made the com- 

 mon vertex of two cones pass- 

 ing respectively through those 

 two curves, the two cones will 

 have the same principal axes 

 and the same cyclic planes. 



5. If a plane be drawn cut- 

 ting an ellipsoid, and a coaxal 

 and biconcyclic ellipsoid be de- 

 scribed touching this plane, 

 the line joining the centre to 

 the point of contact is a prin- 

 cipal axis of the cone having 

 its vertex at the centre, and 

 passing through the plane sec- 

 tion of the first ellipsoid. 



6. And if the other two co- 

 axal and biconcyclic surfaces 

 of the second degree, tangent 

 to that plane, be described, the 

 lines drawn from the centre to 

 their two points of contact with 

 it are the other two principal 

 axes of the same cone. 



7. If a tangent plane to an 

 ellipsoid intersect one of its 

 cyclic cylinders, the cone ha- 

 ving its vertex at the centre 

 of the surface, and passing 

 through the section of the cy- 

 linder has for one of its princi- 

 pal axes the right line drawn 

 to the point of contact of the 

 tangent plane. 



