Roberts — On some Properties of a Certain Quintic Curve. 45 



we may write this in the form 



-u-a^ {N-2I{6)] = N, .... (3) 



the signification of which is that tlie cuhic through the triplets 

 corresponding to u and v passes through the residual of the pair of 

 consecutive points at P. "We can, however, draw the tangent at P 

 by means of two conies as follows — 



Let 



U = Oi + ffo + «3 , ?' = ^1 + So + ^3 5 



then through the points corresponding to the points ai , a^ , ii , h<i 

 describe an conic meeting the quintic again in three points Cj, Ci, Cj. 



Since 2 1{Q) + «< + t; = 0, 



we have 2 /(^) + Ci + Co + Cs + flfa + &3 = iV, 



which proves that the five points Cj , c^_^ c^, a^ , and h^ lie on conic 

 which touches the quintic at P. Hence the tangent required is 

 determined by di-awing the tangent to this conic at P. 



13. — If a line be drawn through one of the Q points to touch the 

 curve in P, then the tangent at P', its corresponding point, will pass 

 through Q' . and the correspondence between the lines QP and QP' 

 will be of the kind noted in Article 10. 



A\, Ai, Ai are the A points lying on a line, and Q and Q' the Q points. 



Now twelve tangents can be drawn from each Q point to the 

 curve, and to each tangent from Q, such as QP, corresponds a tangent 

 Q'P', so that the anharmonic ratio of any four tangents from Q is 

 equal to that of the four corresponding tangents from Q.'. 



K.I. A. PROC, VOL. VUI., SEC. A.] 



E 



