On the Surfaces of the Second Order. 293 



axis in W, the ratio of SF to SD will be expressed by the linear 

 modulus, as will also the ratio of YF to WD, since Y is a point 

 of the surface, and WD is equal to the directive distance of Y 

 from AD. But since Y is a focus to which the mean axis is 

 directrix, the ratio of SY to SW is expressed by the same mo- 

 dulus. Thus the triangles SYF and SWD are similar, the sides 

 of the one being proportional to those of the other. Therefore 

 the angle SYF is equal to the angle SWD ; that is to say, the 

 angle which the side YS of the cone makes with the focal line 

 YF is equal to the angle contained by two right lines WD and 

 WS, of which one is the intersection of the directive plane with 

 the dirigent plane YWD corresponding to YF, and the other is 

 the intersection of the directive plane with the plane YWS 

 passing through the mean axis and the side YS of the cone. 



Hence it appears that the sum of the angles (pspperly 

 reckoned) which any side of the cone makes with its two focal 

 lines is constant. For if F 7 be a point on the other focal line, 

 and D' the point where the directrix corresponding to F' is in- 

 tersected by the same directive plane SWD, it may be shown, as 

 above, that the angle SYF' is equal to the angle SWD', that is, 

 to the angle made by the right line WS with the right line 

 WD', in which the directive plane intersects the dirigent plane 

 corresponding to YF'. Conceiving therefore the points F, F', S, 

 and with them the points D, D', to lie all on the same side of 

 the principal plane which is perpendicular to the internal axis, 

 the right line WS will lie between the right lines WD and WD', 

 and the sum of the angles SYF and SYF' will be equal to the 

 angle DWD', which is a constant angle, being contained by 

 the right lines in which a directive plane intersects the two diri- 

 gent planes of the cone. This constant angle will be found to be 

 equal, as it ought to be, to one of the angles made by the two 

 sides of the cone which are in the plane of the focal lines, namely, 

 to the angle within which the internal axis lies. 



If we conceive the cone to have its vertex at the centre of a 

 sphere, and the points F, F', S to be on the surface of this sphere, 

 the arcs of great circles connecting the point S with each of the 



