530 Proceedings of the Royal Irish Academy. 



It remains to discuss the invariant T= (Ay)^. The yanishing of 

 this invariant may be geometrically expressed in various ways. "When 

 T= 0, the planes through any chord and the four points in which the 

 chords through and 0' respectively meet the curve, form a har- 

 monic system. Again, when T= 0, the polar quadric of passes 

 through 0', and V'ice versa. 



In this investigation I have adopted throughout the notation and 

 procedure of Clebsch, which lends itself more readily than other 

 methods to the identification of binary forms with their geometrical 

 significations. 



KOTE ADDED TK THE PEESS. 



Eor convenience, I give a list of the invariants and covariants of a 

 system of two cubics, of which there are, according to Clebsch and 

 Gordan, twenty-eight forms in all, and which I discuss geometrically. 



Professor Cayley has, however, drawn my attention to the fact 

 that two of the linear covariants {xa^a? and xa^a^) of Clebsch and Gor- 

 dan have been shown to be non-fundamental by Professor Sylvester. 

 See Sylvester's " Tables of the Generating Punctions," Araerican 

 Journal of Mathematics, t. ii. (1879.) 



Table of Covariants and Invariants of a System of Two Ckibics. 



Seven Invariants — 



aa ; «*, a^a, a^ a?, aa?, a* : a^ a?. 



{aay; (AA7, (A0)^ (A^f, (V®)^ (vVT, (®A) (®v) (Ay)- 

 Six Linear Covariants — 



xa'^a, xaa? ; xa^a, xa^ a', xdr a?, xac&. 



(Aa^-a,, (v«r-«.; (aA)^-(aA')A'„ {Q^J Q, {K^fK, («v)H«V') Vx'- 



Six Quadratic Covariants — 



x'^a', x-aa, arar, o?c?a, x-a-ar, x-aa?, 



A,^ ®:-, v.'(A®)A,0., (Av)A,v., (V®)v/.0.- 



Six Cubic Covariants — 



x^a, a? a, a?c^, a?(C'a, x^aar, xr'a?. 



«/, a/, Q.?, (Aa)A.a/, (v«)V.^/, K'- 



One Quartic Covariant — 



x^ aa. 



{a a) a J' a/. 



In this Table I have identified, at the suggestion of Professor Cay- 

 ley, the Covariants given by Sylvester, with the notation of Clebsch 

 and Gordan. 



