OF ARTS AND SCIENCES. 387 



Many other properties and their reciprocals can in the same way be 

 deduced from known properties of plane conies. 



24. The directrix being the conic polar of the focus, its equation ia 



X tan c e 



— ^2— = 1, or-a:=l. 



From this it can be proved that the sines of the distances of any point 

 of the conic from a focus and the corresponding directrix are in a con- 

 stant ratio. By the formulas of Art. 3, if q denotes the distance of 

 any point x'y' of the conic from the directrix, then 



a -p ex' 

 sin n = 1. 



If q' denotes the focal distance of x'y', then 



Sin n' = r ; 



but y'^=^^^(a'-x'% ^ 



1 e2 



1 -|- a-e^ 



, a -F ex' 

 ,*. sm o' ■= — 1, 



. sinj/ _ / a2 ^ £2 



• • ship— y^-qj^- • 



Then, by the method already used several times, we have the recip- 

 rocal : — In a conic, the sine of the angle which a tangent to the curve 

 makes with the cyclic arc is in a constant ratio to the sine of the dis- 

 tance of this tangent arc from the conic jiole of the cyclic arc. 



25, The equation of an arc can be written in the form 



X cos « -|- y sin a = tan p ; 



and from this it can, as in plane conies, be shown that the perpendicu- 

 lar from the centre on a tangent satisfies the equation 



■tan^ p ■= a? cos^ a -\- Ir' sin^ a. 



This property cannot, as in plane conies, be used to find the locus of 

 the intersection of tangents at right angles to each other. The latter 

 problem can best be solved by finding the cone which is the locus of 

 the intersection of tangent planes to a given cone at right angles to 

 each other. 



