Roberts — On Binary Cubics and their Associated Forms. 523 



If we now suppose X=a^, Y=-a2, Z=ai, W= -Uq, tlie equation 

 determining the parameters of the two points in which the chord 

 through 0, the corresponding point of/, meets the curve is — 



{oq tto, - a-c) x^~ + ((?o c-z - di a^) i^i^2 + (^1 «3 - ciz) a^a' = 0. 

 But this is the equation of the Hessian of/, or — 



A/ = Ao ^- + 2 Ai A'l 3^2 + An iTj- . , 



Thus, then, to the Hessian of / correspond the two points in which 

 the chord through meets the curve. 



III. 



I shaU now show that the plane / passes through the line of inter- 

 section of the osculating planes at the two Hessian points. To prove 

 this, let us .find the equation of the plane through and this line. 

 " Let 



X', Y', Z', W ; X", Y", Z", W", 



be the co-ordinates of the two Hessian points, respectively, then the 

 equation of such a plane must be — 



'XiXW -3YZ' + SZr- WX') -ix{XTF" - 3 YZ" + SZY"- WX") = 0, 



where 



fji=aaX' + 8a,Y' + 3a. Z' + a^W. 

 Eemembering that 



Xiyo + 1/1X2 = a^a^i — aitto, 

 Xoyo^aQtto- tti, 



we find, dividing by a factor y^x^ - Xyy.^, that the equation of the 

 sought plane is — 



R{a^X+Za^Y\3a2Z\a3W] =0. 



Hence the plane /passes through the line of intersection of the oscu- 

 lating planes at the Hessian points. If, then, the chord through 

 meet the cubic in real poiats, the plane / must meet it in two ima- 

 ginary points, since the binary cubic is then the difference of two 

 cubes. 



