NEWSON: UNICURSAL CURVES BY METHOD OF INVERSION. 57 



Making these substitutions and reducing we have (dropping the sub- 

 scripts), 



x2-j-2xy-|-y2 x2 — 2xy-|-y2 4aexy (x-j-y) b^x^yS 



a2 ~ b2 ~ a2 ~ ^^ ^ ° 



Making this equation homogeneous by means of z, we have 



( (x2-|-2xy-j--y^) (x^ — 2xy-(-y2) ) 4aexyz(x-|-y) b^x^yS 



{ a3 b2 



=0, 



which is the equation of the quartic referred to the triangle formed by 

 the three nodes. We are now able to determine the nature of the 

 node at the vertex (y, z). Factor x^ out of all the terms which contain 

 it; and arrange thus: 



4aeyz b^y^ ) C yz^ yz^ 2aey2z ) 



+ 2X^ -— -+— ^__ ^^— KJ^ 



y 



( a2 b2 a2 a2 \ ' ) a^ ' b^ 



a2 b2 —^ 



The quantity which multr'plies x^ represents the two tangents at the 

 double point (y, z); but this quantity is a perfect square and hence we 

 have a cusp. In this way the point (x, z) may be shown to be a cusp. 

 Lastly, when a parabola is inverted from the focus, we obtain a tri- 

 cuspidal quartic. 



The trinodal quartic can be generated in a manner analogous to that 

 shown for the nodal cubic. Let two projective pencils of rays have 

 their vertices at A and B, the locus of intersection of corresponding 

 rays is a conic through A and B. Invert from any point O in the plane, 

 and we obtain two systems of coaxial circles, O A being the axis of one 

 and O B of the other. The locus of intersection of corresponding 

 circles is a bicircular quartic having a node at O. Projecting the whole 

 figure we have the following theorem: — two projective systems of conies 

 through O P Q A and O P Q B generate by their corresponding inter- 

 sections a trinodal quartic having its nodes at O, P, and Q, and pass- 

 ing through A and B. 



It is evident that the quartic generated in this way may 

 have three nodes, one node and two cusps, two nodes and one cusp, 

 or three cusps, dependingup on the nature of the conic inverted and 

 the centre of inversion. Making this the basis of classification we thus 

 distinguish four varieties of unicursal quartics. To these must be add- 

 ed a fifth variety, viz: the quartic with a triple point. Each of these 

 varieties will be consided separately. 



The method of treating unicursal quartics given in this and the next 

 four sections is in some respects similar to that suggested by Cayley 



