newson: unicursal curves by method of inversion. 65 



cusps with respect to each of these conies is on the corresponding 

 chord ; the conic through the points of intersection of these two 

 conies and the cusps passes also through the point where the 

 cuspidal tangent of the first mentioned cusp cuts the quartic. Recip- 

 rocating: — on one of the inflectional tangents, of a nodal cubic take 

 two points P and Q; draw a pair of tangents from each of these points 

 to the cubic; draw two conies each touching a pair of these tangents 

 and the other two inflectional tangents, so that the polars of the point 

 of intersection of the other two inflectional tangents with respect to 

 each of those conies pass respectively through P and Q; the conic 

 touching the common tangents to these two conies and the three 

 inflectional tangents touches also the tangent from the first mentioned 

 point of inflection to the cubic. 



QUARTICS WITH A TRIPLE POINT. 



Since a triple point is analytically equivalent to three double points, 

 a quartic with a triple point is unicursal. Such a quartic is obtained 

 by inverting a unicursal cubic from its node. The equation of such a 

 cubic may be written U3-]-U3=o, where Ug and U3 are homogeneous 

 functions of the second and third degree respectively in x and y. Hence 

 the equation of the inverse curve isu3-f-U3(x2-|-y^), which shows that 

 the origin is a triple point and the quartic circular. By projecting this 

 all other forms maybe obtained. 



The nature of the triple point depends upon the relation of the line 

 at infinity to the cubic before inversion. Thus the line at infinity may 

 cut the cubic in three distinct points all real, or one real and two imag- 

 inary, in one real and two coincident points (an ordinary tangent), or 

 in three coincident points (an inflectional tangent). Hence the quartic 

 may have at the triple point three distinct tangents all real, or one real 

 and two imaginary, one real and two coincident, or all coincident. 



This quartic may be generated in a manner similar to that used for 

 the curves already discussed. We showed in the section on nodal cubics 

 that a system of conies through A, B, C, D, and a projective pencil of 

 rays with its vertex at A generate by the intersection of corresponding 

 elements a cubic with a node at A. Invert the whole figure from A and 

 then project: — the pencil of rays remains a pencil; the system of con- 

 ies becomes a system of unicursal cubics having a common node at A 

 and passing through five other common points; the cubic inverts and 

 projects into a quartic with a triple point at A, passing through the 

 five other common points of the system of cubics. 



The three points of inflection of a nodal cubic lie on a right line. 

 Inverting: — there are three points on a circular quartic with a triple 

 point whose osculating circles pass through the triple point, and these 



