[HARDING] RATIONAL PLANE ANHARMONIC CUBICS 18<J 



Since the roots of (28) are equal, it follows that a rational an- 

 harmonic cubic cannot have an ordinary double point. Hence every 



such cubic has one and only one cusp. The cusp parameter is / = — . 



m 



Substituting this value of /in (1) we obtain 



/«QN „ mn8xi= -3{^m-{-yl) {lai-{-2mbi-\-7ic{) .. . 



^^^ n^axi=-3{^m^yl) (lbi + 2mci-^ndù, ^ ' ' ^ 



according as 5 is, or is not, different from zero. Now {^m-\-yiy = l'^n + 0. 

 Hence the coordinates of the cusp of an anharmonic cubic are 



according as ô is, or is not, different from zero. 



The number of points of inflection of a curve of the «th degree 

 is equal to 3«(« — 2) — 6f/ — 8r, where d denotes the number of double 

 points and r the number of cusps. Hence every rational anharmonic 

 cubic has one and only one inflection point. A glance at (19) shows 



that the flex parameter is t= — — . 



After substituting this value of tin (1) it can be easily shown that the 

 coordinates of the point of inflection are 



(30) :x:i = (3TW+5m)aai+(5/3m + T05èi + 262/ci. (i = l, 2, 3) 



6. The tangents from a given point to the cubic. The para- 

 meters of the points of contact of the four tangents which may be 

 drawn from any point x\ to a rational cubic are the roots of the equa- 

 tion* 



(31) \ahx\t^+2\acx\t^+{\adx\^-2,\hcx\)f-^2\hdx\t+\cdx\=0. 

 Since an anharmonic cubic is of class three, this quartic must reduce 

 to a cubic. It can be easily shown that the left member is divisible 

 by (mt — l). Dividing by this factor we obtain 



(32) 



lm\bax\t^-{-l{l\bax\-\-2m\cax\)t^-\-m{2m\bdx\-^n\cdx\)t-\-m'^\cdx\ =0. 

 The roots of this equation are the parameters of the points of contact 

 of the three tangents which can be drawn from any point to the 

 anharmonic cubic. 



If the point X; happens to be on the cubic the tangent at this 

 point counts as two and only one other tangent is possible. We 

 propose now to find the value of the parameter which corresponds to 

 its point of contact. Let x = at^-\-Sbt^-{-3ct-\-d, then 



*J. E. Rowe, Bulletin American Mathematical Society, Vol. 22, No. 2. 



