100 Mr. Lubbock on a Property/ of the Parabola. 



Ingleborough 



Great Whernside 



Whernside (in Ingleton Fells) 



PendleHill 



Boulsworth Hill 



Rumbles Moor 



llkley, May 24, 1836. 



Diff. 



+ 23-5 

 + 57-8 

 + 42-0 

 + 14-7 

 + 19-0 

 + 22-5 



John Nixon. 



XXII. On a Property/ of the Parabola. By J. W. Lubbock, 



'Esq., F.R.S.\ 

 XN the 8th volume of Gergonne's Annates de Mathema- 

 *■ tiqucs, p. 9, M. Poncelet has given the following theorem : 



" Uii triangle etant circonscrit a une parabole, si on lui 

 circonscrit a son tour une circonference de cercle, elle passera 

 necessairement par le foyer nieme de la courbe." 



See also a paper by M. Steiner in the 19th volume of the 

 same work. 



The proofs which have been given of this elegant property 

 of the parabola are indirect, and however ingenious they 

 may be, it seems desirable to show how the theorem in ques- 

 tion may be deduced immediately from the equation to the 

 curve. The general methods of analytical geometry may be 

 deemed incomplete and imperfect while they do not embrace 

 questions of this nature, and their great advantage is liable to 

 be overlooked. 



Let A B C be a triangle, and let a-, , j/, , x<^, y^, x^, j/^, be 

 the coordinates of the points A, B, C. I propose to prove 

 that if the lines A B, B C, A C touch a parabola the focus of 

 the parabola is in the circumference of the circumscribing 

 circle ABC. 



The equation to any straight line passing through given 

 points (j^MJ/i), (-rgji/o) is 



The equation to the tangent passing through the point 

 (x,, i/i) and touching a curve in the point {x, y) is 



d?/ , 



m). 



This equation is generally given for rectangular coordinates 



* This height is probably inclusive of the Beacon hillock. 

 t Communicated by the Author. 



