ELISHA MITCHELL SCIENTIFIC SOCIETY. 15 



ON THE FOCAL CHORD OF A PARABOLA, 



* 



R. H. GRAVES. 



Let y 2 =4ax be the equation to a parabola, S its focus, and 

 PSP r a focal chord. Let the tangent and normal at P' meet the 

 diameter through P at M and N. 



It may be easily proved that PM=PN=PP' and that a simi- 

 lar property holds for the tangent and normal at P. 



Therefore, if two equal rhombs be constructed on PP' having 

 two other sides of each parallel to the axis, their diagonals are 

 tangents and normals at P and P' ; and the tangent at one point 

 is parallel to the normal at the other. 



Each normal chord divides the other in the ratio 1 :3. 



The chord joining the other ends of the normal chords is 

 parallel to PP' and three times as long. 



A line perpendicular to PP r at S, and terminated by this 

 parallel chord and the pole of PP', is divided by S in the ratio 

 1:4. 



Hence the locus of the foot of the perpendicular dropped 

 from S on the parallel chord is a right line, whose equation is 

 x=9a. 



Hence the envelope of the parallel chord is a con focal para- 

 bola, having for its equation y 2 =32a(9a — x). 



It cuts the original parabola orthogonally where it is cut by 

 its evolute. 



*This article has been translated and appeared in the Jornal de Sciencias Mathemat- 

 icas e Astronomicas, published at Coimbra. 



