Section III, 1918 [185] Trans. R.S.C. 



Rational Plane Anharmonic Cuhics. 



A. M. Harding, Ph.D., University of Arkansas. 



Presented by C. T. Sullivan, Ph.D., F.R.S.C. 

 (Read May Meeting, 1918). 



In this paper we propose to study the rational plane cubic curves 

 whose parametric equations are 



(1) >'i = ai/^+36i/2 4-3ci/+<fi, (^"=1,2,3) 



where a,, &,, c-^, d\ are constants. We shall first find the condition that 

 the cubic be an anharmonic cubic. We shall then determine the co- 

 ordinates of the vertices of the invariant triangle, the singular points, 

 and other points connected with an anharmonic cubic. 



It will be convenient to adopt the following notation: 



(2) a=\hcd\, 3i3=|c<fa|, ?>'Y=\dah\, h=\abc\, 



(3) / = /3--a7, 2m = a8-^y, n = y--l38. 

 It is easy to show that the identities 



(4) aia-3M+3ci7-c?i5 = 0, (i=l, 2, 3) 



(5) aw + 2/3m + 7/ = 0, l3n-\-2ym-\-8l = 0, 

 exist among the determinants in (2) and (3). 



For certain values of the constants a, b, c, d, equations (1) may 

 represent a straight line or a conic. In particular, if a = /3 = 7 = ô = 0, 

 the coordinates yi, yo, ys are linearly dependent and the locus is a 

 straight line. We shall assume at the outset that a, /3, 7, ô are not 

 all zero. 



1. The differential equation. Following Wilczynski we shall 

 study the anharmonic cubic by means of a linear homogeneous 

 differential equation of the third order, of which yi, y^, yz are solutions. 

 This differential equation may be written 



(6) 3''" + 3/>i3'"+3p23''+/>33' = 0, 



where pi, pi, pz are functions of t. It is easily shown that, if yi, y^, ys 

 are solutions of (6), we must have 



(7) -A^=/3+27^+5/2, 



A/,2 = 2(7+0/), 



where 



