newson: unicursal curves by method of inversion. 49 



nodal curics. 



If an ellipse be inverted from one of its vertices, the inverse curve 

 is symmetrical with respect to the axis; it has one point of inflection 

 at infinity and the asymptote is an inflectional tangent. This asymp- 

 tote is the inverse of the circle of curvature at the vertex. The cubic 

 has two other points of inflection situated symmetrically with respect 

 to the axis. Hence the three points of inflection lie on a right line, 

 a projective U^eore-m which is consequently true of all nodal cubics. 

 The axis is evidently the harmonic polar of the point of inflection at 

 infinity. Since the axis bisects the angle between the tangents at the 

 node, it follows that the line joining a point of inflection to the node, 

 the two tangents at the node, and the harmonic polar of the point of 

 inflection, form a harmonic pencil. There are three such lines, one to 

 each node, and three harmonic polars; these form a pencil in involution, 

 the tangents at the node being the foci. 



Since the asymptote is perpendicular to the axis, we have by projec- 

 tion the following theorem: — through a point of inflection I, draw any 

 line cutting the cubic in B and C. Through P the point of intersection 

 of the harmonic polar and inflectional tangent of I, draw two lines to 

 B and C. The four lines meeting in P form a harmonic pencil. The 

 point of contact of the tangent from I to the cubic is on the harmonic 

 polar of L Any two inflectional tangents meet on the harmonic polar 

 of the third point of inflection. 



The locus of the foot of the perpendicular from the focus of a conic 

 on a tangent is the auxiliary circle. Inverting from the vertex, there 

 are two points, A and B, on the axis of the curve, such that if a circle 

 be drawn through one of them and the node, cutting at right angles a 

 tangent circle through the node, their point of intersection will be on 

 the tangent to the curve where it is cut by the axis. Projecting: — 

 through a point of inflection I of a nodal cubic draw a line cutting the 

 cubic in P and Q ; there are two determinate points on the harmonic 

 polar of I, which have the following property: — draw a conic through 

 P, Q, and the node touching the cubic; draw another conic through 

 one of these points, P, Q, and the node cutting the former, so that 

 their tangents at their point of intersection, together with the lines 

 from it to P and Q form a harmonic pencil; the locus of such a point 

 of intersection is the tangent from I to the cubic. 



If three conies circumscribe the same quadrilateral, the common 

 tangent to any two is cut harmonically by the third. Inverting from 

 one of the vertices of the quadrilateral: if three nodal, circular cubics 

 have a common double point and pass through three other fixed 

 points, the common tangent circle through the common node to any 

 two of the cubics is cut harmonically by the third; /. c, so that the 



