479 



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 per- 

 pendicular to the side VS 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 VSF 

 and VSF', because the right line FE is perpendicular to the 

 plane VSF, and the right line F'E' is perpendicular to the 

 plane VSF'. Tlierefore the tangent plane of a cone makes 

 equal angles with the planes drawn through the side of con- 

 tact 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 ^ (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 So, 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 S E (or of FS' to S'E) is to the ratio of FSo to SoE 

 (or of FSi to SiE) as the value of in for the one surface is to 

 its value for the other, when the focus is modular, or as the 

 value of /x for the one surface is to its value for the other 

 when the focus is umbilicar, the siues of the angles EFSo 

 and EFS (or of the angles EFS, and EFS') are in a con- 

 stant proportion to each other, because these sines are pro- 



