323 
349] ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 
and consequently 
that is 
X + y + Z = /3y {/3 - 7), 
x : y : z : — x + y + z : x — y+z:x+y— z 
= “ 2 (£ ~ 7) : /3 2 (7 ~ a ) : 7 2 (« ~ 0) ■ fiy (@- 7) : 7« (7 ~ a ) : «/3 (a - /3), 
and these give 
{— x + y + z) {x —y + z) {x + y - z) + xyz = 0, 
or, what is the same thing, 
x 3 + y 3 + z 3 — {yz 2 + ?/ 2 ^ + .sa? 8 + z l x + xy- + x-y) + = 0, 
as the equation of the locus of the one-with-twofold centre, which locus is thus a cubic 
curve. 
29. The equation 
x? + y s + z 3 — (yz 2 + y‘Z -1- zx} + z 2 x + xy- + x-y) + 3xyz = 0, 
of the one-with-twofold centre locus may be transformed as follows, viz. writing for a 
moment x + y + z= — w, we have 
(9x + 4w) (9y + 4w) (9z + 4w) — w 3 
= 729 xyz + 324 w {yz + zx + xy) — 144 w 3 + 64w 3 — w 3 , 
= 81 {9xyz + 4w {yz + zx + xy) — w 3 ), 
= 81 {9xyz — 12xyz — 4 {yz- + &c.) + {x :i + y 3 + z 3 ) +• (3yz 2 4- «See.) + 6xyz], 
= 81 [a? + y 3 + z 3 — {yz 2 + &c.) + Sxyz], 
so that the equation may be written 
(9# + 4w) (9y + 4w) (9z + 4iu) — w 3 — 0, 
or, what is the same thing, 
{5x — 4y — 4z) (— 4x + by — 4z) {— 4# — 4y + 5z) + {x + y + z) 3 = 0, 
which shows that the intersections of the line x + y + z — 0, with the sides x = 0, y = 0, z = 0 
of the triangle are inflexions on the curve; and that the tangents at these points are 
respectively 
5x — 4y — 4<z = 0, — 4« + 5y — 4<z = 0, — 4« — 4y + oz = 0. 
30. The curve passes through the point (1, 1, 1), which is the harmonic of x + y + z = 0 
in regard to the triangle; and this point is moreover a node on the curve; in fact 
if the equation be represented by W = 0, then we have 
d x W = Sx- — 2x {y + z) — y- + Syz — z-, 
= 0 for the point in question; and similarly d y W and c4TF=0. 
41—2