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points F and A, and let AD be the directrix passing through 

 A. Then if a directive plane, drawn through any point S 

 of the surface, cut this directrix in D and the mean axis in 

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

 modulus, as will also the ratio of VF to WD, since V is a 

 point of the surface, and WD is equal to the directive dis- 

 tance of V from AD. But since V is a focus to which the 

 mean axis is directrix, the ratio ofSV to SW is expressed by 

 the same modulus. Thus the triangles SVF and S WD are simi- 

 lar, the sides of the one being proportional to those of the other. 

 Therefore the angle SVF is equal to the angle SWD ; that 

 is to say, the angle which the side VS of the cone makes 

 with the focal line VF 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 VWD corres- 

 ponding to VF, and the other is the intersection of the di- 

 rective plane with the plane V WS passing through the mean 

 axis and the side VS of the cone. 



Hence it appears that the sum of the angles (properly 

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

 focal lines is constant. For if F' be a point on the other 

 focal line, and D' the point where the directrix correspond- 

 ing to F' is intersected by the same directive plane SWD, 

 it may be shown as above that the angle SVF' 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 in- 

 tersects the dirigent plane corresponding to VF'. Conceiv- 

 ing therefore the points F, F', S, and with them the points 

 D, D', to lie all on the same side of the principal plane 

 wliich is perpendicular to the internal axis, the right 

 hne WS will he between the right lines WD and WD', and 

 the sum of the angles SVF and SVF' 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 

 dirigent planes of the cone. This constant angle will be 



