278 On the Stirfaces of the Second Order. 



and hence the equation (35) becomes 



(x z - x,} (x - a?i) + (y z - y,) (y - yj = 0, (37) 



which, as is evident from (33), is the equation of a tangent 

 applied to the focal at the point F corresponding to A. This 

 shows that the focal and dirigent are reciprocal polars with 

 respect to the section xy, and that in this relation, as well as in 

 the other, the points F and A are corresponding points. 



Supposing F' and A' to be two other corresponding points 

 on the focal and dirigent, if tangents applied to the focal at F 

 and F' intersect each other in T, the point T will be the pole of 

 the right line A A' with respect to the section xy, as well as the 

 pole of the right line FF' with respect to the focal ; and hence 

 if any right line be drawn through T, and if P be the pole of 

 this right line with respect to the section, and N its pole with 

 respect to the focal, the points P and N will be on the right 

 lines A A' and FF' respectively. Now it is useful to observe 

 that the distances A A' and FF' are always similarly divided 

 (both of them internally or both of them externally) by the 

 points P and N, so that we have AP to AT as FN to F'N. 

 This property may be proved directly by means of the fore- 

 going equations ; or it may be regarded as a consequence of the 

 following theorem : If through a fixed point in the plane of 

 two given conies having the same centre, or of two given para- 

 bolas having their axes parallel, any pair of right lines be 

 drawn, and their poles be taken with respect to each curve, 

 the distance between the poles relative to one curve will be in 

 a constant ratio to the distance between the poles relative to 

 the other curve.* In fact, the poles of the right lines TF, TF', 

 with respect to the focal, are F, F' ; and their poles with respect 

 to the section xy are A, A'; therefore, since the focal and the 

 section xy may be taken for the given curves, and the point T 



* There is an analogous theorem for two surf aces of the second order which 

 have the same centre, or two paraboloids which have their axes parallel. If 

 through a fixed right line any two planes be drawn, and their poles be taken with 

 respect to each surface, the distance between the poles relative to the one surface 

 will be in a constant ratio to the distance between the poles relative to the other. 



