349] 
ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 
317 
13. Considering the critic centre corresponding to a root d of the cubic equation, 
the equation of the line joining the other two critic centres is 
\x ixy vz 
1—U-z—| — () 
d + A d + /x 0 + v 5 
which is the polar of the critic centre in regard to the twofold centre conic. The 
critic centres are consequently conjugate poles in regard to the twofold centre conic. 
14. The equation of the tangents at the critic centre considered as a node of the 
corresponding cubic curve is 
(o + 4A, 0 + 4<y, 0 + 4>v, - 6 - , - 8 - ~ , - 6 - (x, y, z) 2 = 0. 
15. The last-mentioned formulas lead to some which involve the three critic centres 
viz. if X = 0, Y =0, Z = 0 are the equations of the sides of the triangle formed by 
the critic centres, then the equations of the tangents at the three critic centres 
respectively are of the form 
BY' 1 + CZ' 1 — 0, 
AX 2 . aGZ 2 =0, 
AX 2 + BY 2 . =0, 
so that the tangents in question are in fact the tangents from the three nodes 
respectively to the conic 
AX 2 + BY 3 + CZ 2 = 0: 
the three nodes or critic centres being thus conjugate poles in regard to the conic, 
this is called “ the three centre conic.” 
16. The equation of a nodal cubic is also expressible in a simple form in terms 
of the new coordinates X, Y, Z. In the formulae for these transformations, and indeed 
throughout the memoir, the three roots of the equation in 6 are represented by 
di, d 2> d 3 , and I write also 
¿i = d 2 — d 3 , L=0 3 - d l5 l 3 = 0 1 — d 2 , 
= ($i + V) (^1 + y) (0i + v), 
0 2 = (d 2 + A) (d 2 + ¡x) (d 2 + v), 
= (d 3 + A) (d 3 + y) (d 3 + v). 
17. If Aa + nb+ vc = 0, that is, if (a, b, c) are the coordinates of a point on the 
line + ¡xy + vz = 0, then the critic centres lie on the cubic 
a b c 2(ct + b + c) 
x y z x + y +z 
or, what is the same thing, this curve is the locus of the critic centres corresponding 
to the several lines \x + /xy + vz =■ 0 through the point (a, b, c).