1893.] Covariants of Plane Curves, fyc. 425 



^ 2 U 7 U 9 -Y 8 2 -4U 7 3 + ^V 8 = o. 



Tbe matrix of the equation to the tangents to the cubic from the 

 origin is 



(0 + 0iL 6 + 2 L 6 2 ) 2 - 4 / ( yjr + Y^Le) , 



and the condition that the cubic may be nodal or cuspidal is that 

 this matrix, as a function of L 6 , may have a linear factor twice or 

 thrice repeated. 



In the case of the nodal cubic, the differential equation will be 

 derived from U 7 y<- = V 8 /, and, putting \> = 2fk, h is a root of 



U 7 Z; 4 -l^ + 2U 7 2 Z: 2 -|w 5 4 U 7 Z; + U 7 3 + H2i 5 8 = 0, 



and from this the differential equation is found. 



In the case of the cuspidal cubic, put ^ = 2/&, w/0 = fq. 



Then 16 A; 3 = 27 ti 5 \ 



236 (f — 27u 5 *, 



and the differential equation is 



256 U 7 3 — 27uf = 0. 



§5. 



In this section it is attempted to develop a geometrical method, 

 founded on the covariant theory. 



The general equation to a covariant line takes the form 



(«5-0^4+0i + 02L 6 — 03^) 77— usih { (02 — 2 (fijbi) — u& 6 as } w 3 3 % 3 3 



= 0, 



and depends upon the invariant ratios t : 2 : b . 



If we take a second covariant line, the coordinates of the point of 

 intersection take the form 



7T = w 2 3 A/w 4 A-f- C +Bcij, 



where 



A = — U 5 2 (0 2 0' 3 — 0' 2 03), 



B = «- 5 { (030'l-0 / 30l) ~ (020' 3 - 0' 2 3 ) Lj}, 



C = (0 1 0' 2 -0' 1 2 )+2(0 3 0' 1 -0' 3 1 ) L 6 -(0 2 0' 3 -^3)L3 2 . 



Or, more shortly, 



A = ti 5 2 X, 



B = W 5 (/LI — \L 6 ), 



C = ^ + 2 /( L,-AL/~, 



VOL. T.I1I. 2 H 



