158 Messrs. W. B. Morton and T. C. Tobin on the 



from a point A on the curve to any other two points B, P, 

 and if the diameters through B, P are drawn to intersect AP, 

 AB respectively, then the line joining the points of inter- 

 section is parallel to the tangent at A. 



Fig. 1. 



Referring to fig. 2 in which the parabola is placed in the 

 more usual posture, assume that the points A, B and the 

 tangent at A are given, and construct P in the manner 

 indicated above. It can easily be shown that P lies on the 

 curve. 



For AN : AK = PN : HK = PN : BM 



and AK : AM = IK : BM = PN : BM 



.-. AN : AM = PN 2 : BM 2 . 



It is interesting to notice that this property is really a 

 special case of Pascal's theorem about a hexagon inscribed 

 in a conic *. Let the angular points 1, 2, of the hexagon be 

 at A, 3 at B> 4, 5 at the point at infinity on the axis of the 

 parabola, and 6 at P. Then side 12 is the tangent at A, 

 23 is AB, 34 is BH, 45 is the line at infinity which touches 



* This way of looking at the matter was pointed oat to us by 

 Mr. F. M. Saxelby. 



