newson: unicursal curves bv method of inversion. 51 



and the point at infinity is the point of inflection. The directrix of 

 the parabola inverts into a circle through the cusp of the cissoid 

 having the cuspidal tangent for a diameter. Hall calls this the 

 directrix circle. The double ordinate of the parabola which is tangent 

 to the circle of curvature of the vertex inverts into the circle usually 

 ■'called the base circle of the cissoid.* 



The cissoid may fairly be called the simplest form of the cuspidal 

 cubic. Its projection and polar reciprocal are both cuspidal cubics. 

 I shall now deduce from the parabola a few simple propositions for 

 the cissoid, and then extend them to all cuspidal cubics. 



(i) It is known that the locus of the intersection of tangents to the 

 parabola which are at right angles to one another, is the directrix. 

 Inverting: — the locus of the intersection of tangent circles to the 

 cissoid through the cusp and at right angles to each other is the 

 directrix circle. 



(2) For the parabola, two right lines O P and O Q, are drawn 

 through the vertex of the parabola at right angles to one another, 

 meeting the curve in P and Q; the line P Q cuts the axis at a fixed 

 point, whose abscissa is equal to its ordinate. Inverting: — two right 

 lines, O P and O Q, are drawn at right angles to one another 

 through the cusp of the cissoid, meeting the curve in P and Q; the 

 circle O P Q passes through the intersection of the axis and asymptote. 



(3) If the normals at the points P, O, R, of a parabola meet at a 

 point, the circle through P O R will pass through the vertex Invert- 

 ing: — -through a fixed point and the cusp of a cissoid, three and only 

 three circles can be passed, cutting the cissoid at right angles; these 

 three points of intersection are collinear. 



From the geometry of the cissoid we see that if any line be drawn 

 parallel to the asymptote, cutting the curve in two points, B and C, 

 the segment B C is bisected by the axis. Hence, projecting the curve 

 we have the following theorem: — any line drawn through the point of 

 inflection is cut harmonically by the point of inflection, the curve, and 

 the cuspidal tangent. Thus the cuspidal tangent is the harmonic 

 polar of the point of inflection. The polar reciprocal of this last 

 theorem reads as follows: — if from any point on the cuspidal tangent 

 the two other tangent lines be drawn to the curve, and a line to the 

 point of inflection, these four lines form a harmonic pencil. These 

 are fundamental propositions in the theory of cuspidal cubics. 



(4) Projecting proposition (i) above, we have the generalized theo- 

 rem: — through the point of inflection draw any line cutting the cubic 

 in B and C; through B, C, and the cusp draAv two conies tangent to 



*See note B. 



