. JotY—Vecfdr Expresswiisfur Curves'. 389 



14. Syzygy of points, curvesj and planes. 



Take for example a conic. Let the point x-^y^ on it be denoted by 

 the symbol (11), and the tangent line thereat by the symbol (1). A 

 point on this tangent line may be denoted by the symbol. (12) or (^1),' 

 and the second tangent through this point may be denoted by (2), and 

 its point of contact by (22). 



Again, for a cubic, the first or conic emanant at x^yi may be sym- 

 bolized by (1), the tangent line at the point by (11), the osculating 

 plane by [1], and the point itself by (111). The point whose para- 

 meter is x^ : yt on the conic (1) may be called (122), and the tangent 

 line thereat (12) or (21). In general, the order in which the figures 

 occur within the brackets is arbitrary. 



Two figures complete a syzygy for a conic, consisting of two points 

 (11) and (22) on the conic, their pole (12), and the tangents (1) and (2). 



15. Description of a syzygy for the twisted cubic. 



For a cubic a complete syzygy of points, curves, and planes may be 

 derived from three figures. In the osculating plane [l] lie the points, 

 lines, and the conic involving the figure 1 in their symbol. The planes 

 [1] and [2] intersect in the line (12). The lines of intersection of the 

 three osculating planes [1], [2], and [3] are (23), (31), and (12), and 

 they intersect in the point (123). This point has been called in Art. 6 

 the pole of the three points (111), (222), and (333). 



In the plane [1] are the lines (11), (12), and (13), and these are 

 tangents at the points (111), (122), and (133) to the conic (1). The 

 points (122) and (133) are the points in which the tangents (22) and 

 (33) to the cubic meet the plane [1]. But since the lines joining the 

 points of contact of a conic inscribed in a triangle to the opposite ver- 

 tices concur, the lines joining (111) to (123), (122) to (113), and(133) 

 to (112) concur in some point Pi. If P2 and P3 are points similarly 

 formed in the planes [2] and [3], the following groups of collineations 

 may be written down : — 



(HI), (123), P,; (122), (113), P,; (133), (112), P,; 



(222), (123), P,- (233), (221), P,; (211), (223), P,; 



(333), (123), P3; (311),(332), P,; (322), (331), P,. 



Again, taking a plane through thepoints (113), (221), and (332) ; 

 in virtue of the collineations it passes through Pj, Pj, and P3. In like 



