4 260 



XIII. As the tangent to a parabola makes equal angles with the 

 focal radius vector and a perpendicular on the directrix, so also 

 this tangent makes equal angles with the normals to the Logocyclic 

 curve drawn through this point of contact. 



XIV. The two reciprocal points R and R x on the Logocyclic, the 

 logarithmic point Q on the parabola, and the intersection V of the 

 tangents to the Logocyclic which meet on the Cissoid lie on the 

 circumference of a circle, or in other words, any two reciprocal 

 points on the curve, the intersection of the normals at these points 

 and the intersection of the tangents at the same points lie in the 

 circumference of a circle. 



XV. Through the two reciprocal points R and R x a circle may 

 be drawn touching the axis at O, and having its centre at T the 

 intersection of the vertical tangent OT, with the radius vector FR. 

 This property leads to a simple construction of the curve by points. 



XVI. The tangent to the curve at either of the reciprocal points 

 makes with the radius vector through this point an angle whose tan- 

 gent is equal to the cosine of the inclination of the radius vector to 

 the axis. 



XVII. The sum of the polar sub tangents FC and FC, belonging 

 to the reciprocal points R and R, is constant, and equal to 2a. 



XVIII. The sum of the reciprocals of the polar subnormals be- 



2 

 longing to two reciprocal points is constant, and equal to - 



XIX. The lengths of the tangents to the curve between any 

 two reciprocal points and the asymptote are equal, or R*=R/,, and 

 S*=S^=a secfl. 



XX. The products of these two tangents t, and the perpen- 

 diculars p,p t from the focus F upon them, is constant, or t*pp t =a 4 . 



XXI. Let p and p t be the reciprocals of the radii of curvature of 

 the Logocyclic curve at the points R and R, ; and let p fi be the 

 reciprocal of the radius of curvature of the parabola at a point 

 through which the normal makes the angle \fr with the axis, then 



xi/ and are connected by the condition tan ^=cos 6. See (XV.) 



XXII. The Logocyclic curve is the envelope of all the circles 

 whose centres range along the parabola, and whose radii are succes- 

 sively equal to ^f 2 ', /being the distance of the centre, Q, of the 



