Iharding] rational PLANE ANHARMONIC CUBICS 191 



where 



(37) z = y'+p,y, 



^ ^ P = y"+2p,y'+p,y. 



The geometric significance of the points defined by the fundamental 

 covariants (36) has been explained by the author in a former paper.* 

 Any other covariant may be expressed in terms of these three co- 

 variants and of invariants. 



From (1), (7), (19), we find, after reduction 



(38) 

 mKz = Same -^md-\- (Qamb - 3ylc-\-8ld)t-{- iSama + S^mh - 6ylb)t^ 



-^{2l3ma-3yla-3ôlb)t^-2ôlat\ 

 {l-mt)Kp 



/ 



Whence 

 (39) 



XCi = 3amc-{-yld-2l3md-\-3(2amb-^mc-\-ôld)ti'3(ama 



= 6ab -6^c-\-2yd-\-8aat+ l2^at- + 8yat^-{-28at\ 



60il3m-j-yl)l -ylb-^2ôlc)r-+{l3ma-2yla-\-3ôlb)t\ 



-^— — X Cg = VZa'^nb - 18al3nc + Gaymc + 3^^nd-\-3aynd - 2a8md 

 3600/^ _4^^^^ 



-h{16a''na-\-S6aymb-\-24aôlb-l5^8lc-{-6aômc-SyHc 



- lO08md-\- I0y8ld)t 



+ (13a5/a+ 15ai8«a+ l8a8mb-30-nb-\- 18^ymb-9yHb 



-l2^8mc-\-88Hd)t^ 



■i- (27 ^8la-\- 21 ay na-\-lOa8ma-\-2Q0yma-\-G(38mb 



- I87Ô/6+ 128Hc)t\ 

 The fact that the right members of these two equations are cubics 

 agrees with a theorem proved by the author in a former paper, namely, 

 that every covariant point either remains fixed, describes a straight 

 line, or describes a curve which is projective with a given turve. 



9. The invariant triangle. Associated with every anhar- 

 monic curve of the first class is a triangle, called the invariant triangle, 

 which has the following property. Suppose any tangent is drawn to 

 the curve. The anharmonic ratio of the point of contact and the 

 three points of intersection of this tangent with the three sides of the 

 triangle is constant for all points on the curve. The vertices of this 

 triangle are the only covariant points whose coordinates reduce to 

 constants. They are defined by the covariant j 



*Harding, Giornale di Matematiche, VoL 54, No. 3. 

 fHarding, Giornale, VoL 54, No. 4. 



