6o KANSAS UNIVERSITY QUARTERLY. 



Salmon). Hence its reciprocal is also a nodal bicuspidal quartic, a 

 fact of which frequent note will be made in this section. 



The inverse of a conic with respect to a focus is a curve called 

 Pascal's Limacon. From the polar equation of a conic, the focus 

 being the pole, it is evident that the polar equation of the limacon 

 may be written in the form: 



r= — cos,x -\ : 



P P 



where e and p are constants, being respectively the eccentricity and 

 semi-latus rectum of the conic. 



From the above equation it is readily seen that the curve may be 

 traced by drawing from a fixed point O on a circle any number of 

 chords and laying off a constant length on each of these lines, meas- 

 ured from the circumference of the circle. The point O is the node 

 of the limacon; and the fixed circle, which I shall call the base circle, 

 is the inverse of the directrix of the conic. This is readily shown as 



follows: — the polar equation of the directrix is r= — - — . Hence the 



ecos^r 



CCOSiX 



equation of its inverse is r= , which is the equation of the base 



P 

 circle of the lima9on. 



The envelope of circles on the focal radii of a conic as diameters is 

 the auxiliary circle. Inverting: — the envelope of perpendiculars at the 

 extremities of the nodal radii of a limacon is a circle with its centre 

 on the axis and having double contact with the limacon. Projecting: — 

 from any point on a nodal bicuspidal quartic draw lines to the three 

 nodes and a fourth line forming with them a harmonic pencil; the en- 

 velope of all such lines is a conic through the two cusps and having 

 double contact with the quartic; the chord of contact passes through 

 the node and cuts the line joining the cusps so that this point of inter- 

 section, the two cusps, and intersection of the double tangent with the 

 cuspidal line form a harmonic range. Reciprocating: — on any tan- 

 gent to a nodal bicircular quartic take the three points where it cuts 

 the two inflectional tangents and the double tangent, and a fourth point 

 forming with these a harmonic range; the locus of all such points is a 

 conic touching the two inflectional tangents and having double contact 

 with the quartic; the pole of the chord of contact is on the double 

 tangent; join this last point to the intersection of the inflectional tan- 

 gents and join the node with the same intersection; these four lines 

 form a harmonious pencil. 



If the tangent at any point P of a conic meet the directrix in Q, the 

 line P Q will subtend a right angle at the focus O; the circle P O Q has 



