98 Proceedings of the Rnj/al Irish Academjj. 



If these pass througli a point S'\>x^S<^i'^B<^i^ = 0. This is the Jacobian of the 

 three quadrics. 



Consider the cubic Spi\, (pp) = 0. Put p = p + x-u, we have 



•Spi// (pp) + SxSa'ip (pp) + 3:i:-Sptp (<j<r) + X^Sa-^ ((T(t) = 0. 



If we regard p as fixed, this gives us the intercepts on the line p + xrr, drawn 

 from p II (T to meet the cubic, so that the properties of cubics may be 

 investigated. 



Sp-ip (<t(t) = may be called the polar cone, and S(7-ip (pp) = 0, the polar 

 plane of tr with respect to the cubic. 



If the equation in a; has two equal roots, we get a condition for o- 



4 [Sp^P (o-(t) Sp-p, (pp) - SgxP [ppf] [S(TiP (aa) SmP (pp) - Spxp (o-a)'] 



= [Sinp (pp) Sp-ip ((T(r) - Sp\p (pp) Strip (cro-)]-, 



a cone of 6th degree in a. If Spxp (aa) = breaks up into two planes, 

 the line of intersection is along p. In this case its discriminant vanishes, or 



S^ip(pipi\)zp = = Srpi(Tip2<J(j>:iT. 



Since ^ (ap) = -ip (pa) = (A.). Thus the Hessian of 



Sp\p (pp) = is Sfipip.p^^p = ; 



and i// ipa) = gives the relation between corresponding points. This 

 relation may be written 



Vcp^p(p3P _ F(^3pj>lP _ V(plp<P2f> 



SXcT Sfiu Sva 



If Sp(pi<T = = Spip2T = Sptpzrr, each of these vectors is parallel to a. 

 p and o- may be interchanged in these formulae. 



II. 



10. The general quadratic vector function may be expressed in terms 

 of 3 vectors oi, oo, as, the coefficients being scalar quadratic functions of p. 

 If we denote it by x (pp). 



•^ (pp) = aiSp(pip + aiSp(p2p + asSpf^p. 



Hence it contains in general 18 constants, there being no loss of generality 

 in supposing the <^'s self-conjugate. 



11. The as and the "^'s determine the function. Suppose a's transformed 

 by the substitution 



a'l = ^"u"! + /'"ijos + ^1303. (§ 7), 



