I 



477 



of both, the sum or difference of the distances SF and SF' 

 will be constant. 



If the plane of section pass through either of the foci, as 

 F, this focus and its directrix will manifestly be the focus 

 and directrix of the section. In this case the plane of section 

 will be perpendicular to the focal at F. And if the surface 

 be a cone, the point F being anywhere on one of its focal lines, 

 the distance of the point S from the directrix will be in a 

 constant ratio to its perpendicular distance from the dirigent 

 j)lane which contains the directrix, and therefore this per- 

 pendicular distance will be in a given ratio to the distance 

 SF. Now calling V the vertex of the cone, and taking SV 

 for radius, the perpendicular distance aforesaid is the sine 

 of the angle which the side SV of the cone makes with the 

 dirigent plane ; and SF, which is perpendicular to VF, is 

 the sine of the angle SVF\ Consequently the sines of the 

 angles which any side of a cone makes with a dirigent 

 plane and the corresponding focal line are in a given ratio to 

 each other. 



§ 2. Conceive a surface of the second order to be inter- 

 sected in two points S, S' by a right line which cuts two 

 parallel directrices in the points E, E', and let F, F' be the 

 foci corresponding respectively to these directrices. The 

 perpendicular distances of the points S, S' from the first di- 

 rectrix and from the second are to each other as the lengths 

 SE, S'E, SE', S'E' respectively, and therefore the ratios of 

 FS to SE, of FS' to S'E, of F'S to SE', and of F'S' 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 F'E' bi- 

 sects 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' ap- 

 proach to it indefinitely, the angle SEE will approach inde- 



