524 Proceedings of the Royal Irish Academy. 



lY. 



Let us now determine the co-efficients of tlie plane 



X{XW' -3YZ'+ ZZY' - WX') + fjL{XJF" - 3 YZ" + ZZY" - WX") = 0, 



and we find, after a few obvious reductions, that it becomes — 



^{{aottz - Saoaia-z + 2a^)X+S{aoaias + ttiUo - 2aaCio^) Y+ 3{2ai^a3 - at^a^a^ 

 ' - axa-i^Z-\- (3aia2fi'3 - fl'o a^ - 2a^) W] = 0, 



but this is the cubic covariant plane Q. 



"We see, then, that the planes /and Q are harmonically conjugate 

 with regard to the osculating planes at the Hessian points, and that 

 since the Hessian of the cubic Q^ is the same as that of «/, the cor- 

 responding point of the plane Q lies on the chord through 0, and is 

 the harmonic conjugate of with regard to the Hessian points. 

 Hence, to the cubic covariant corresponds the plane through the inter- 

 sections of the osculating planes at the Hessian points and the har- 

 monic conjugate on the chord through of the same point with regard 

 to the Hessian points. Again, if the point lie at one side of the 

 developable generated by tangent lines to the cubic curve from which 

 a real chord can be drawn, two of the roots of the binary cubic are 

 imaginary ; if the point lies on the developable, two roots are 

 equal ; and if at the other side, from which a real chord cannot be 

 drawn, all the roots are real. 



Y* 



"We can now discuss the system of two binary cubics and their 

 associated forms, and shall adopt the notation of Clebsch in our inves- 

 tigation. Let then / and <^ denote the two cubics, A and v their 

 Hessians, Q and ^ their cubic covariants, and R and P their discrimi- 

 nants, f "We have then 



f = a^x^ + 3ai x^Xi + 3«2^i ^2^ + ttzX:i = a^ = h^ = c^ = &c. ; 

 <f) = o-qX-^ + Sai^^i^ifs + SaoS^-a^o^ + a^X'? = a/ - j3,f = y/ = &C. 



There is one quartic form, (ffa) a^a^, which we shall first discuss. 

 The co-ordinates of the line of intersection of the planes / and ^ are, 



h = 3 (flfias - ttop-i), g= 3 {ao,ao - aoo^), 



c-3{aoa3-a3a2), h-3(aoai- aio^j), 



* See Note at end of this Paper. 



t The reader is referred to Clebsch's TJieorie der Bindren Algebraischen Formcny 

 § 61. Vollstandiges System zweier cubischen Formen. 



