390 Proceedings of the Royal Irish Academy. 



manner, a plane tLrough (112), (223), and (331) passes througli Pi, 

 P-i, and P3 ; and since these planes do not in general coincide, Pi, P2, 

 and P3 must lie on a line. Hence it follows that (111), (222), (333), 

 and (123) lie in a plane, as has been more generally proved for curves 

 of odd order in the 7th article.^ 



16. Linear construction for this syztjgy. 



The properties proved in the last article give a means of construct- 

 ing, not only the conies, but the three points of osculation, when the 

 osculating planes and tlie tangents are given. 



The intersections of the planes determine the lines (23), (31), and 

 (12). In the plane [1], the five points (112), (113), (122), (123), and 

 (133) arc given, since they are points of intersection of the given and 

 constructed lines. The point Pi is constructed by joining (122) to (113), 

 and (133) to (112) ; and the point (111) lies on the line joining Pi to 

 (123). The conic (1) in this plane is uniquely determined, as it has 

 to touch the three lines (11), (12), and (13) at the constructed points 

 (111), (112), and (113). 



It should be remembered that it has been proved, in Art. 13, that 

 these conies lie on the tangent-line developable of the cubic. The 

 theorem respecting the locus of their centres, given in Salmon's " Three 

 Dimensions," will be generalized in a future article of the present 

 Paper.'' 



17. Syzygy for the twisted quartic. 

 The syzygy for the twisted quartic 



_ (g^^aia oaaaj) {xy)*' 



^ ~ («o«i«2«3«ij {xyY 



consists of the following system : — Denoting a point on the curve by 

 (1111), the first emanant (a twisted cubic) at this point by (1), the 

 second emanant (a conic) by (11), the tangent line by (HI), and the 

 osculating plane by [11] ; there are four sets of points, cubics, conies, 

 lines, and planes, whose symbols involve only one of the four figures 

 1, 2, 3, and 4. In addition, there are the mixed emanant conies (12), 

 and their planes [12]. The conic (12) may be described either as the 



' See Art. 337 of Dr. Salmon's "Three Dimensions." 



2 See " Three Dimensions," Art. 340 ; and Art. 21 of this Paper. 



