THE NORMAL TO THE CONIC SECTION. 



Edmund A. Engler. 



The determination by the usual methods of analytic geom- 

 etry of the normals to a conic section from a point not on the 

 curve involves the solution of a cubic or bi-quadratic equa- 

 tion by which the co-ordinates of the points at which the 

 normals cross the curve may be found. This method, 

 though of no special difficulty, is generally avoided on account 

 of its tediousness, and in most elementary books the problem 

 is considered solved when these equations are found. The 

 graphical constructions given in this paper are accomplished 

 without the algebraic solution of these equations and may 

 prove to be a useful addition to courses in geometrical draw- 

 ing, from which the solution of this problem is generally 

 omitted. The analysis on which the constructions depend is 

 based on well known principles of analytic geometry and is 

 given in the most elementary and simple form. 



THE NORMAL TO THE PARABOLA. 



The equation of the parabola referred to its axis and the 

 tangent at its vertex as axes of co-ordinates is, in the usual 

 notation, 



y 2 = 2px. ( 1) 



The equation of the normal to the parabola is 



py + %Vi — pVi + ^iVi ( 2 ) 



in which x v y x are the co-ordinates of the point in which the 

 normal crosses the curve. 



(137) 



