14 Proceedings of the Royal Irish Academy. 



If three lines are in perspective with their derived lines, and if a is 

 the axis of perspective, the lines lie on the cone Sp(f>ipa = 0. This 

 furnishes an interpretation of the vanishing of the skew scalar inva- 

 riant JVs'-JVa, in Art. 19 of the Paper on Scalar Invariants. In the 

 notation of that article, if pi, pj, and ps are the axes of <f>, and if 



6pi = Unpi + «12P2 + <?13P3) &C. , Fpi^pi = ff 12 Fpipz + «!i3 FpiPj, 



and 



8 VpiOpi VpiOpi VpsOpi = («12<1!23«'31 - «21«532«'l3) ^ ^zPs ^sPl ^PlPs- 



The skew invariant consequently vanishes, if the derived lines of the 

 axes of either vector function with respect to the other function are 

 in perspective with them. 



Finally, it may be noticed, that if jS is an arbitrary vector, the 

 locus of axes of the functions ^ip + aS^p is the quadric cone Spcf)ipa = 0. 



19. — Co-residual property. — The axis of the functions ^i + t(p2 are 

 all co-residual triads of lines on the cubic cone. Por the four lines 

 common to the cone 



SaVp{cf)i + t(f>2)p = 



(which contains the axes of <^i + tcf^i), and to the cone Sa Vpcft^p = 0, 

 satisfy the equation 



xa = V. Vpcfiip Vp (<^i + tcl>^) p 

 = - tpSpcfiipfji^p. 



Three of the lines are therefore on the cone Spcjiiptji^p = 0, and are 

 variable with a, but independent of t. The fourth line is, of course, a. 

 The three lines common to the cubic, and Sapcfi^p = are therefore 

 residual to every triad of axes. 



If the elliptic parameters of the axes of «^i + t(j>2 are Wi, u^, and «,, 

 their sum is constant, or 



Unless, therefore, u^ is equal to half Si period, the axes of two func- 

 tions ^1 + i5^2 an4 «^i + t'<i>.i will not lie on a quadric cone. 

 Again, the four lines common to the quadric cones satisfy 



xp = V. Fa(^ip Fa^2P 



= - aSa(f>ip^2P- 



Consequently, the three edges which lie on the cubic lie also on the 

 new quadric 



Sacfiipcfizp = 0. 



