478 



finitely to a right angle. Therefore if a right line touching the 

 surface meeta directrix in a certain point, the distance between 

 this point and the point of contact will subtend a right angle 

 at the focus which corresponds to the directrix. And if a cone 

 circumscribing the surface have its vertex in a directrix, the 

 curve of contact will be in a plane drawn through the cor- 

 responding focus at right angles to the right line which joins 

 that focus with the vertex. 



When the surface intersected by the right line SS' is a 

 cone, suppose this Hne to lie in the plane of the focus F and 

 its directrix, that is, in the plane which is perpendicular at 

 F to the focal line VF (the vertex of the cone being de- 

 noted, as before, by V) ; the angles made by the right lines 

 FEjFS, FS', are then the same as the angles made by planes 

 drawn through YF and each of the right lines \'E, VS, 

 VS' ; and the last three right lines are the intersections of 

 a plane VSS' with the dirigent plane on which the point E 

 lies, and with the surface of the cone. Therefore if a plane 

 passing through the vertex of a cone intersect its surface in 

 two rightlines, and one of its dirigent planes in another right 

 Hue, and if a plane be drawn through each of these right 

 lines respectively and the focal line which belongs to the di- 

 rigent plane, the last of the three planes so drawn will bisect 

 one of the angles made by the other two. And hence, if a 

 plane touching a cone along one of its sides intersect a diri- 

 gent plane in a certain right line, and if through this right 

 line and the side of contact two planes be drawn intersecting 

 each other in the focal line which corresponds to the dirigent 

 plane, the two planes so drawn will be at right angles to each 

 other. 



Let a right line touching a surface of the second order 

 in S meet two parallel directrices in the points E, E', and let 

 F, F' be the corresponding foci. Then the triangles FSE 

 and F'SE' are similar, because the angles at F and F' are right 

 angles, and the ratio of FS to SE is the same as the ratio of F'S 



