[HARDING] R.\TIONAL PLANE ANHARMONIC CUBICS 187 



But if l = m = 0, if follows from (5) that either a = j8 = or n = 0. 

 If a = /3 = it follows from (3) and the last equation of (13) that n = 0. 

 Hence a necessary and sufficient condition that equations (1) repre- 

 sent a conic is l = m = n = 0. It is interesting to note that in this case 

 Pi also vanishes. We shall assume throughout this paper that one of 

 the determinants /, m, n, is different from zero. For convenience we 

 shall first assume that neither /, m, nor n is equal to zero. The special 

 cases (1) l = m = 0, w + 0, and (2) / + 0, w = w = will be considered 

 later. 



4. The condition that equations (1) shall represent an 

 ANHARMONIC CUBIC. The invariant 6» is defined by the equation 



(14) ds = Qdsd/ -7{dsy-27P2e3\ 



It is easily seen that, after the substitution of P2 and ^3 in this equation, 

 one would obtain A^ds — R^, where R^ is a rational function of / of at 

 most the eighth degree. 



Equations (1) will represent an anharmonic cubic if, and only if, 



In this case 



(R^y = constant X (^ + Wt-h'SCt^-{-Dt;'y. 

 Hence, a necessary condition that the cubic (1) be anharmonic is that 

 the expression A-^-SBt-^-SCf^-hDt^ be of the form Mfx-^-vty, where 

 X, n, V are constants. 

 The cubic 



(15) A-\-SBt-\-3Ct~-hDtr' = 

 will have three equal roots if, and only if, 



B--AC = AD-BC=a-BD=0. 

 From (12), (3), (5), we find 



B' -AC = l(ln -m2), 



(16) AD-BC=2m(ln-m^), 

 a -BD=n{ln -m"). 



Hence the cubic (15) will have three equal roots if 



(17) m^-ln = 0. 

 From (16) it follows that 



{AD-BCy-4{B^~-AC) (O - BD) = 4{ln - m^Y . 

 Since the left member of this equation is the discriminant of the cubic 

 (15), the cubic cannot have two equal roots without being a perfect 

 cube. 



Making use of (5), (17), we obtain certain identities which will be 

 of use in the sequel. 



