On the Surfaces of the Second Order. 291 



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 FS to SE'. 

 Therefore the tangent EE' makes equal angles with the right 

 lines drawn from the point of contact S to the foci F, F'. When 

 the surface is a cone, let the tangent be perpendicular to the 

 side YS which passes through the point of contact ; the angles 

 FSE and F'SE' are then the angles which the tangent plane 

 VEE' makes with the planes YSF and YSF', because the right 

 line FE is perpendicular to the plane YSF, and the right line 

 F'E' is perpendicular to the plane YSF'. Therefore the tangent 

 plane of a cone makes equal angles with the planes drawn 

 through the side of contact and each of the focal lines. 



Supposing a section to be made in a surface of the second 

 order by a plane which cuts any directrix in the point E, if the 

 focus F belonging to this directrix be the vertex of a cone having 

 the section for its base, the right line FE will be an axis of the 

 cone. For if through FE any plane be drawn cutting the base 

 of the cone in the points S, S', one of the angles made by the 

 sides FS, FS' which pass through these points will always be 

 bisected by the right line FE ; and this is the characteristic 

 property of an axis. 



3. Two surfaces of the second order being supposed to 

 have the same focus, directrix, and directive planes, so that they 

 differ only in the value of the modulus m, or of the umbilicar 

 ratio fj. (see Part I. 9) : let a right line passing through any 

 point E of the directrix cut one surface in the points S, S', and 

 the other in the points S , Si, and conceive right lines to be 

 drawn from all these points to the common focus F. Since, if 

 ratios be expressed by numbers, the ratio of FS to SE (or of 

 FS' to S'E) is to the ratio of FS to*S E (or of F& to S X E) as 

 the value of m for the one surface is to its value for the other, 

 when the fociis is modular, or as the value of ju for the one sur- 

 face is to its value for the other when the focus is umbilicar, the 



