JoLY — Vector Expressions for Curves. 391 



conic emanant at the point (1222) on tlie cubic (1), or as that at (2111) 

 on the cubic (2). This conic (12) is related to the conies (11) and (22) 

 as follows. The planes [11] and [12] intersect in the line (112) which 

 touches the conic (11) at (1122), and also the conic (12) at (1112). 

 Similarly, the line (122) lies in the planes [22] and [12], and this 

 line touches (22) at (1122), and (12) at (1222). The line of inter- 

 section of [11] and [22] cannot be expressed by a symbol of the kind 

 here used, but (1122) is a point on it. The point (1122) lies on each 

 of the conies (11) and (22), and the plane [12] touches both the conies 

 at this point, as it contains the tangent to each. Again, this point 

 (1122) is the pole of the chord joining (1112) and (2212), two points 

 on the conic (12). These points lie on tangents to the quartic, and 

 generally (12) meets the tangents (111) and (222), tangents to the 

 quartic, and to the conies (11) and (22) respectively. 



Again, for three figures, there is the line (123), through which the 

 planes [23], [31], and [12] pass, and which is a tangent to the three 

 conies (23), (31), and (12) at (1123), (2231), and (3312), respectively. 

 Similarly, introducing a fourth figure, three new lines (234), (314), 

 and (124) are found, and these lines intersect with (123) in the point 

 (1234), which is the pole of the four assumed points. Through this 

 pole pass the six planes of the type [12], which intersect by threes in 

 the lines of the type (123). 



18. Remarks on the general sy%ygy. 



In general, for a curve of the w*^ degree, the pole of n points 

 Xiyi; x^y-i'i . . . ^„y„ may be denoted by the symbol (1, 2, . . . »). 

 Through this point pass ^n {n-\) planes of the type [1,2,.,. (w- 2)], 

 whose symbols involve only (» - 2) of the n figures. These planes 

 intersect in n lines (1, 2, . . . (?z- 1) ), through each of which n-\ 

 planes pass. Given n -\ points, and combining them with an arbi- 

 trary w"* point on the curve, the locus of the poles is the line 

 (1, 2, . . .{n—V)). Given only (?i-2) points, and combining them 

 with two arbitrary points, the locus of the poles is the plane 

 [1, 2, . . . (w-2)] ; but if the same arbitrary point is taken twice 

 over, the locus is the conic (1, 2, . . . (?^ - 2) ). In general, the ema 

 nant curves may be considered as loci of poles. Thus the first emanant 

 (1) is the locus of the poles of the system consisting of a given point 

 iCi^i, and an arbitrary point xy taken n-\ times. 



R.I A. PKOC, SEE. UI., VOL. IV. 2 E 



