349] ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 351
80. The line 3x + y + z = 0 is the line through the points (—1, 4, —1), (—1, — 1, 4),
and as such the corresponding critic centres lie
one on the line z + x = 0, two on the conic x(x + y + z) — tyz = 0,
one on the line x + y = 0, two on the conic y {x + y + z) — 4zx = 0.
The two lines meet in the point (1, — 1, — 1).
The two conics meet in the points (1, 0, 0), (2, 3, 3); and touch at the point
(0, 1, —1), the common tangent being 5x + y + z = 0: this appears by writing the
equations of the two conics in the forms
(y -z)(5x + y + z) + (y + z)(-3x + y + z) = 0,
-(y-z)(5x + y + z) + (y + z)(-3x + y +z) = 0,
for we have then the four points of intersection put in evidence; viz. these are
y — z = 0, y + z = 0, that is (1, 0, 0),
y-z = 0, - 3x + y + z = Q, „ (2, 3, 3),
5x + y + z = 0, y + z = 0, „ (0, 1, -1),
5x + y + z = 0, -3x + y + z = 0, „ (0, 1, - 1).
The point of intersection (1, 0, 0), which is an angle of the triangle, is not a critic
centre; the three critic centres are the other point of intersection (2, 3, 3); the point
of contact (0, 1, —1); and the point of intersection (1, —1, —1) of the two lines.
81. To obtain in a different manner the last-mentioned result it may be remarked
that for the line 3x + y + z = 0, for which (X, y, v) = (3, 1, 1), the equation in 0 is
6 s - 70 - 6 = (0 +1) (0 + 2) (0 - 3) = 0,
so that the values of 0 + X, 0 + y, 0 + v are
for d = -l, 2, 0, 0,
„ 6 — — 2, 1, -1, -1,
„ 0= 3, 6, 4, 4,
and the corresponding values of x : y : z are
i :
go :
oo, that is, (0,
1,
-1),
1 :
-1 :
- 1, „ (1,
-1,
-1),
i .
F •
i ■■
b * (2,
3,
3),
which points are therefore the critic centres for the line 3x + y + z = 0.
The last-mentioned line, it is clear, is one of the system of three lines
3x + y + z = 0, x + 3y + z = 0, x + y + 3z = 0.
82. If X = 0, that is if the line \x + yy + vz = 0 pass through an angle y = 0, z — 0
of the triangle; then reverting to the original equations
— x + y + z_x — y + z_x + y — z
\x yy
vz
