190 THE ROYAL SOCIETY OF CANADA 



\bax\ = \b, a, Sct-\-d\ =3|èac|/+ \bad\ = —3ôt — 3y, 

 \cax\ = \c, a, 3bi'^+d\ =3\cab\t^--\-\cad\ =S8t^ -Sfi, 

 \bdx\ = \b, d, af + 3ct\ = \bda\t^-\-3\bdc\t = 3yt^-3at, 

 \cdx\ = \c, d, at^+Sbf^l = \cda\t^-]-3\cdb\t^=S^t^-\-Sat\ 



Substituting in (32) we obtain, after reduction, 



[-m-\-y)T-\-m{^t-\-a)] {T-ty = 0, 



where we have replaced / in (32) by T. Hence the tangent to the cubic, 



drawn from any point on the curve, touches the cubic at the point 



defined by the parameter value 



(33) r = ^x^'+^ 



7. The Halphen Point. Solving (33) for /, we obtain 



ylT—am 



or, after interchanging T and t, 



(34) T= -TliZ^, 



Ut-^m 



That is, the tangent to the curve at any point / cuts the curve again at 

 the point T, where T is defined by (34). This second point has been 

 called the tangential of the first point. 



The system of cubics which have eight consecutive points in 

 common with the given cubic at the point P, all pass through a ninth 

 point which is on the given cubic. Wilczynski has called this point the 

 Halphen point of P. 



Let us regard (34) as a transformation which transforms the 

 parameter of the point P into the parameter of its tangential. If this 

 transformation be twice repeated we obtain 



/oc\ rp_(5yl-{-2^m)t — Sam 



5^m -\-2y I -3ôlt 

 This is the parameter value which corresponds to the Halphen point.* 



8. The fundamental covariants. All points of the plane, 

 which are intrinsically connected with the given cubic, must be 

 independent of the choice of the parameter in (1). That is they must 

 be defined by covariants. The three simplest covariants are 



Co=y, 

 (36) C, = d,'y-\-39^z, 



^8= m'y-\-9P2ds'']y-^Qdzd3'z+ ISdhp, 



*Wilczynski, Projective Differential Geometry, p. 69. 



