443 



this conic, and having its vertex at the focus ; the cyclic planes 

 of this latter cone will pass respectively through the two right 

 lines in which the planes of the two sections meet the directive 

 plane. 



" 5. Hence, through a right line, situated in a directive 

 plane, draw two planes cutting the surface in two conies ; a 

 cone passing through these two curves intersects that directive 

 plane in a certain conic : now let a cone be described, passing 

 through this conic, and having its vertex at the focus ; this 

 latter cone will be one of revolution, and its principal plane 

 will pass through the right line assumed in the directive plane. 



" 6. Hence, a cone enveloping the surface intersects a di- 

 rective plane in a certain conic ; a cone passing through this 

 conic, and having its vertex at the focus, is one of revolution ; 

 and its principal plane passes through the right line in which 

 the plane of contact of the enveloping cone meets the direc- 

 tive plane. 



" 7. A plane curve is traced on the surface, and through it 

 a cone is described, having its vertex at the pole of a directive 

 plane ; this cone cuts the directive plane in a certain conic : 

 now let a second cone be described, passing through this conic, 

 and having its vertex at the focus ; the latter cone is one of re- 

 volution, and its principal plane passes through the right line 

 in which the plane of the original curve intersects the directive 

 plane. 



" 8. If a cone be described, having its vertex at the focus, 

 and passing through a plane section of the surface, the cyclic 

 planes of this cone will pass respectively through the two 

 right lines in which the plane of the section meets the two 

 directive planes ; and the directive axis of the cone will there- 

 fore be the right line drawn from its vertex to the point where 

 the directrix is cut by the plane of the section. 



" 9. Hence, any plane passing through the directrix inter- 

 sects the surface in a curve such that the cone passing through 

 it, and havina: its vertex at the focus, is one of revolution ; 



