demo7istrated by Mr. Lubbock in Phil. Mag. for August. 35 



ing the chord through the focus in Q ; Q O perpendicular to 

 P Q and meeting the normal in O: then O is the centre of 

 curvature. 



The other theorem I found in some manuscripts of the same 

 period, but it was probably never published. It is as follows : 

 If a sphere be described round the vertex of a cone as cen- 

 tre, the latera recta of all sections of the cone made by planes 

 touching the sphere will be constant, and equal to the radius 

 of the sphere multiplied by the tangent of half the vertical an- 

 gle of the cone. I will not add the proofs of these theorems, 

 as they are attended with no difficulty. 



The insertion of this notice will much oblige yours, &c., 



Trinity College, Cambridge. G. 



IX. Concise Demonstration of the Property ^f the Parabola 

 in the Phil, Mag. for August, By the Rev. Hamnett 

 HoLDiTCH, Fellow of Caius College, Cambridge. 



To the Editors of the Philosophical Magazine and Journal, 



Gentlemen, 



TN the last [August] Number of your Magazine is an elabo- 

 ■■■ rate and ingenious demonstration of the theorem thus stated 

 by M. Poncelet, " Un triangle etant circonscrit a une parabole, 

 si on lui circonscrit a son tour une circonference de cercle, 

 elle passera necessairement par le foyer meme de la courbe." 

 I venture to propose for insertion the following very short de- 

 monstration of this theorem : Let (^t'l 3/i)j (^23/2)5 (•^'.sj/a) ^^ 

 the points of the parabola to which the tangents are drawn ; 

 (X12 Y12) the point where the tangents to the points i^x-^y^ 

 {x^ 1/2) "^^^^ 5 ^h^" ^^ y^ — ^ ^^ ^^ ^^^ equation to the parabola, 

 the equation to the tangent at the first point is 



23/3/1 = Ix + Ixy .-. 2 Y123/1 = /X12 + Ix^ 



and 2 Yj2J/2 = /X,2 + /.r^; 

 and we have therefore 



X -y^y^ and Y '-yi±2b 



A.J2 — — ^ ana 1 12 — ^ 



Let x'^^y^-\'ax-\-hy-\-c=^0 be the equation to the 

 circle passing through the three points, 



.-. X^2 + Y^2 + « X12 + 6 Y12 + C = 



and X^23 + Y^ga + « X23 + 6 Y23 + c = ; 

 F2 



