of a Conic Section. 187 



is perpendicular to AX. From P draw PR perpendicular to 

 RX, and therefore parallel to ^^' or JX; and through f"' draw 

 f^F parallel to J.4' or PR to meet EB (produced if necessary) 

 in F, and join FD, DR. Then, because FD, and DR are parts 

 of the same common section, (viz. the common section of the 

 plane BDE with the plane of the parallels VF, PR,) FDR is 

 a straight, line. Lastly, join SP. 



Now, the section of the sphere made by the plane f^'PS 

 is a circle ; and the straight line FP touches this circle in the 

 point D, because it lies in the same plane with it, and meets 

 it in that point only ; and for the like reason PS touches it in 

 the point S; therefore PD is equal to PS. 



Again, because the triangles PDR, VDF are similar, PD is 

 to PR as FD to VF ■, but VD is equal to FB, and it has 

 been shewn that PD is equal to PS; therefore PS is to PR as 

 VB to VF, that is, in a constant ratio. 

 Therefore, &c. Q. e. D. 



It need scarcely be added that the point S thus found is 

 \\\e focus, and RX ihe directrix of the conic section. 



If the plane of the conic section be not parallel to a slant 

 side of the cone, that is, if the section be an ellipse or an 

 hyperbola, a second sphere may be inscribed in the lower 

 segment of the cone (see tig. 1.) or in the vertical cone (see fig. 3.), 

 which shall touch the plane AQP in a second point S', and 

 the conical surface in a second circle, with regard to which 

 the same property obtains. For to this the foregoing demon- 

 stration is equally applicable, the letters with dashes being 

 substituted for the others, each for each. Thus we arrive at 

 the other focus and the other directrix, and in these ca.ses it 

 is easy to perceive that AS is equal to A'S', and AX to A'X. 

 The next proposition will shew how readily the inscribing of 

 these two spheres leads to the simple properties by which the 

 ,ellipse and hyperbola are usually detined. 



A A 2 



