14 JOURNAL OF THE 



ON THE CHORD COMMON TO A PARABOLA 

 AND THE CIRCLE OF CURVATURE AT 



ANY POINT.* 



R. H. GRAVES. 



It is known that if a circle meet a parabola in four points the 

 sum of the distances of the points on one side of the axis from 

 it is equal to the sum of the distances of the points on the other 

 side from it. If three of the points are coincident, the circle 

 becomes the circle of curvature, and the distance of the three 

 coincident points (P) from the axis is one-third of that of the 

 fourth point from the axis. 



Hence the common chord of the circle and parabola is divided 

 by the axis in the ratio 1 : 3. But the shorter segment of the 

 chord is equal to the tangent at P, since they are equally inclined 

 to the axis. Therefore the chord is equal to four times the tan- 

 gent. Let y 2 =4ax be the equation to the parabala, and (x', y') 

 the co-ordinates of P. Then 



y— y'=— p (x— x'), or yy' + 2ax— fy ,2 =o, 

 is the equation to the chord. 



Differentiating with respect to y', y=3y / ; hence y 2 = — 12ax 

 is the envelope of the chord. Also, from the relation y=3y r , 

 it follows that the longer segment of the chord is equal to the 

 corresponding tangent of the parabola y 2 = — 12ax. 



The point P, and the point where the chord prolonged touches 

 y 2 = — 12ax, are harmonic conjugates with respect to the points 

 where it meets the axis and the tangent at the common vertex of 

 the parabolas. 



The tangent at the end of the lotus rectum of y 2 = — 12ax is 

 normal to y 2 =4ax at the end of its lotus rectum, and therefore 

 touches its evolute. The chord is then a diameter of the circle 

 of curvature, and is bisected by its point of contact with the 

 evolute. 



Hence the radius of curvature— twice the normal=4aj 2, 

 which agrees with a known result. 



*This article and the following one have appeared in the ''Annals of Mathematics." 



