Dr Taylor, Geometrical Notes on Theorems, etc. 153 



Geometrical Notes on Theorems of Halley and Frdgier. By 

 C. Taylor,. D.D., Master of St John's College.' 



The two subjects of these notes are Halley's construction 

 for the normals or normal from a given point to a parabola by 

 means of a circle, and Fregier's theorem that a chord of a conic 

 which subtends a right angle at a fixed point of the curve passes 

 through a fixed point on the normal thereat. 



A. 



Halley. 



1. Apollonius shewed how to draw the normals to a conic 

 from a given point by means of a rectangular hyperbola. 



This cuts the conic in four or fewer real points, each lying on 

 a normal to the conic from the given point. 



In the course of his investigation Apollonius found the 

 coordinates of what we call the centre of curvature, and thus 

 virtually the equation of the evolute of a conic. 



From the given point H in certain positions normals can be 

 drawn to a central conic at four points P, p, Q, a. 



When the conic is a parabola one of the four points, as q, is at 

 infinity, and the remaining three, P, p, Q lie on a circle through 

 the vertex A. Halley shewed how to draw the normals HP, 

 Up, HQ by means of this circle. 



2. Draw the ordinates PN, pn, QM, supposing Q to be either 

 of the two points on the parabola having the same abscissa AM 

 and such that 



QM = PN + pn. 



The chords A Q, pP make equal angles with the axis, and in 

 this section they may be parallel. 



Taking the upper sign, draw Ppl to the axis, and let 4>a be 

 the latus rectum. 



Then AM = IN + In = AN + An + 2AI. 



Writing (PN +pn) 2 /4<a for AM and subtracting AN + An 

 from both sides, we get 



PN.pn = 4<a.AI. 



