Mr. Lubbock on some Problems in Analytical Geometry. 85 



and eliminating A, 



(m'^— n^) xy—n? ay + 7i^/3a; = 

 which is the equation to an equilateral hyperbola (see fig. 1.), 

 which cuts the ellipse in the points to which normals can be 

 drawn from the point a, |S. The asymptotes to the hyper- 



bola are parallel to the principal axes of the curve; it passes 

 through the centre and the point (a, ^) ; the coordinates of 



its centre are -?^ and - -?— -7. When a or /3 = 0, 



the hyperbola merges into two straight lines at right angles 

 to each other ; and when the point (a, /3) coincides with the 

 centre, these lines become the principal axes of the curve. 



If the equation to the curve be j/- = 2px, the equation to 

 the corresponding hyperbola is 



xy — {°^-p)y + P/3 = 0. 



The centre is situated on the axis of the parabola at a distance 



a—p from the vertex. This construction is given by Apol- 



loniijs. See Bossut, Histoire des Mathanatigues, vol. i. p. 37. 



If a/ and y' be the coordinates of any point in the normal, 



xf—a, _ x—t*. 



and eliminating x and y between this equation and the equa- 

 tions 



w^'x* + TO^y = m^n^; {m^-n'')xy-vi'ay -f n'-^x = 0, 

 we have 



which is the equation to the normals drawn from any point 



a/3. 



When m = 11, the ellipse becomes a circle: m tins case 



