290 0)1 the Surfaces of the Second Order. 



respectively, and therefore the ratios of FS to SB, of FS' to S'E, 

 of FS to SE', and of FS' to S'E', are all equal. 



Hence, the right line FE bisects one of the angles made by 

 the right lines FS and FS' ; and the right line FE' bisects one 

 of the angles made by F'S and F'S'. 



When the points S, S' are at the same side of E, the angle 

 supplemental to SFS' is that which is bisected by the right line 

 FE. Now if the point S be fixed, and S' approach to it inde- 

 finitely, the angle SFE will approach indefinitely to a right 

 angle. Therefore if a right line touching the surface meet a 

 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 circumscrib- 

 ing the surface have its vertex in a directrix, the curve of contact 

 will be in a plane drawn through the corresponding 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 line to lie in the plane of the focus F and its direc- 

 trix, that is, in the plane which is perpendicular at F to the focal 

 line YF (the vertex of the cone being denoted, as before, by V) ; 

 the angles made by the right lines FE, FS, FS', are then the 

 same as the angles made by planes drawn through YF and each 

 of the right lines YE, YS, YS' ; and the last three right lines are 

 the intersections of a plane YSS' 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 sur- 

 face in two right lines, and one of its dirigent planes in another 

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

 lines respectively and the focal line which belongs to the dirigent 

 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 touch- 

 ing a cone along one of its sides intersect a dirigent plane in a 

 certain right line, and if through this right line and the side of 

 contact, respectively, 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. 



