newson: unicursal curves by method of inversion. 6i 



PQfor a diameter and hence cuts the conic at P at right angles. 

 Inverting: — from any point P on the limacon draw O P to the node O; 

 draw O Q perpendicular to O P meeting the base circle in Q; P Q is 

 normal to the limacon at P. Projecting: — from any point P on a 

 nodal bicuspidal quartic draw lines to the three nodes and a fourth 

 harmonic to these three; from O draw lines to the two cusps and a 

 fourth harmonic to these two and the line O P; the locus of the inter- 

 section of the fourth line of each pencil is a conic through the three 

 nodes. Call this the basal conic of the quartic. Reciprocating: — on 

 any given tangent to a nodal bicuspidal quartic take its points of 

 intersection with the double tangent and the inflectional tangents, and 

 a fourth point harmonic with these; on the double tangent take its 

 points of intersection with the given tangent and the inflectional tan- 

 gents, and a fourth point harmonic with these; the envelope of the 

 lines joining the fourth point of these two ranges is a conic touching 

 the double and inflectional tangents. 



The locus of the foot of the perpendicular from the focus on the 

 tangent to a conic is the auxiliary circle. Inverting: — draw a circle 

 through the node tangent to a lima9on; draw the diameter O P of this 

 circle; the locus of P is a circle having double contact with the lima- 

 9on, the axis being the chord of contact. Cor.; the locus of the 

 centre of the tangent circle is also a circle. Projecting:— through the 

 three nodes of a nodal bicuspidal quartic draw any conic touching the 

 quartic; the locus of the pole with respect to this conic of the line 

 joining the two cusps is a conic; draw the chord O P of the first conic 

 through the node O and the pole of the line joining the two cusps; 

 the locus of P is a conic through the cusps, having double contact 

 with the quartic. 



If chords of a conic subtend a constant angle at the focus, the 

 tangents at the ends of the chords will meet on a fixed conic, and the 

 chords will envelope another fixed conic; both these conies will have 

 the same focus and directrix as the given conic. Inverting: — draw 

 two nodal radii of a lima9on O P and O Q, making a given angle at O; 

 the envelope of the circle P O Q is another limacon; the locus of the 

 intersection of circles through O tangent to the limacon at P and Q is 

 another limacon. These two limacons have the same node and base 

 circle as the given one. Projecting: — through the node O of a nodal 

 bicuspidal quartic draw a pencil of radii in involution; let O P and O Q 

 be a conjugate pair of these nodal radii; the envelope of the conic 

 through P, Q, and the three nodes, is another quartic of the same 

 kind: also draw conies through the three nodes tangent to the quartic 

 at P and Q; the locus of their point of intersection is another quartic 



