465 



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 Nwill be on the right lines A A' and FF' respectively. 

 Now it is useful to observe that the distances AA' 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 A'P as FN to F'N. This property may be 

 proved directly by means of the foregoing equations; or it 

 may be regarded as a consequence of the following theo- 

 rem : — If through a fixed point in the plane of two given 

 conies having the same centre, or of two given parabolas 

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

 and their poles be taken with respect to each curve, the dis- 

 tance 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 for the fixed point, the ratio of 

 FF' to A A' is the same as the ratio of FN to AP or of F'N 

 to A'P, and consequently the distances FF' and AA' are si- 

 milarly divided in the points N and P. 



§9. In the equation (30), considered as equivalent to 

 the equation (1), the constants l and m are both positive ; 

 but the properties which have been deduced from the 

 former equation are independent of this circumstance, and 



• There is an analogous theorem for two surfaces 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 th« distance between the poles relative to 

 the other. 



