Mr. Lubbock on a Property of the Parabola. 101 



only, but the reasoning by which it is established is equally 

 applicable to coordinates inclined to each other at any angle. 



Let y^ = 2px be the equation to a parabola referred to 

 any coordinuate axes Ox, Oy oblique or rectangular. 

 y = Vq, X = FY, p = 2SF. (See Bridge's Conic Sec- 

 tions, p. 15.) 



^ =z P- = y^zy^ 



dx y i-j — Xg' 



if die tangent passes through {x^^y^, {x^, y^. 



y^ y-- 2 {x,y,-x^y,) ^''^ 



Similarly, 



y^ y^- 2 (x,y,~x,y,) ^''> 



y^ y^- 2 {x,y,-x,y,) ^^'^ 



By making the diameter of the parabola O x pass through 

 the point A, I may hereafter make j/j = 0, without limiting 

 the generality of the question. Subtracting (2.) from (1.); 



^1 (i/3-3/2)(3/3+3/2)+i/l (^lJ/2--^2J/> -■^li/3 + ^3i/0 

 J) 



= "^ (2 J^i ^2 "•^3) ("*'3 •^2) 

 pl^x^ + x^) (0-3-^:2) {x^yi-Xc^y-i-x^y^-x^y.^) 



= 23/1 (a:2 J/3— ^33/2) (^i3'2—^2J/i—^i3/3 + ^s^i) (*•) 



Hence if j/i = 



x^ + Xo = or Xq — OTj = 

 or ^33/2-^23/3 + J:,j/3--^i 3/2 = 0. 

 In the second case y^—y^, = 0; and since ^"3 = Xg, t/g = 1/2 

 the points {xc^,y,^, (-^s 5 3/3)9 coincide and are identical; in 

 the third case the points (.r, , j/,), {x^, y^, (■^'s, j/3) are in the 

 same straight line ; it is useless therefore to consider these cases, 

 and it is sufficient to take the first case only, namely, when 

 _y, = 0, and x._^ + x^ = 0. 



Let Xy Y be the coordinates of the centre and R the ra- 

 dius of the circumscribing circle passing through the points 

 (4^1 » yi)i {^'•2i]/<i) i^sij/i)^ then by the equation to the circle, 

 {x^-Xr+(y,-Y)^ + 2{x^-X) iy,-Y) cosxy = W 

 {x^-Xf+{y,-Yf^2 {x,-X) {y,- Y) cos xy = B?. 



