600
PROBLEMS AND SOLUTIONS.
[383
The points of inflexion lie (as is known) by threes on twelve lines; viz., the lines are
abc,
afh,
h/9>
cfi,
adg,
bdi,
cdh,
def,
ae i,
be h,
ceg,
ghi.
Consider now a point on the curve, the coordinates whereof are (x, y, z), where of
course x 3 4- y 3 + z 3 + Qlxyz = 0 ; this is one of a system of eighteen points on the curve,
which may be represented as follows:
A =
(x,
y>
z),
D--
= (
<»y,
arz),
G =
= (
X,
™-y>
ft) z),
B =
(y>
z,
x),
E =
= (® y>
arz,
x),
H =
' % y>
ft) z,
x),
G =
(z,
X,
y)>
F =
- (<*> 2 Z,
X,
™y\
I =
: (tO Z,
X,
rfy)’
J =
(x,
Z,
y\
M =
a,
O) Z,
<* 2 y)>
P =
(
X,
arz,
" y\
K=
{z,
y>
x),
N=
s (a>z,
rfy,
x),
Q =
(û)
2 Z,
® y»
x),
L =
(y>
X,
z),
0 =
-- {<» 2 y,
X,
O) z),
B =
(œ
y>
x,
arz),
the coordinates
of
A
are
(®. y>
z) ; those
of B
are (y,
z,
x),
&c.
The tangent at A meets the curve in a point, “ the tangential of A,” the coordi
nates whereof are x(y 3 — z 3 ), y (z 3 — y 3 ), z (x 3 — y 3 ); which point may be called A'. And
we have thus the eighteen tangentials
A', B', O', D', E', F', G', H', T, J', K', L\ M, N', O', P', Q', R.
The eighteen points A, B, &c., have the following property; viz., the line joining any
two of them meets the cubic in a third point, which is either one of the nine points
of inflexion, or one of the eighteen tangentials; there are through each point of inflexion
9 such lines, and through each tangential 4 such lines; (9 x 9) + (18 x 4)=153 = ^(18.17),
the number of pairs of points AB, AG, &c. The lines through the inflexions are the
81 lines obtained by joining any one of the points {A, B, C, D, E, F, G, H, I) with
any one of the points (J, K, L, M, N, 0, P, Q, B), as shown in the following Table:
A
B
C
D
E
F
G
II
I
J
a
d
9
c
f
i
b
e
h
K
d
9
a
f
i
c
e
h
b
L
9
a
d
i
c
f
h
b
e
M
c
f
i
b
e
h
a
d
9
N
f
i
c
e
h
b
d
9
a
0
i
c
f
h
b
e
9
a
d
P
b
e
h
a
d
9
c
f
i
Q
e
h
b
d
9
a
f
i
c
R
h
b
e
9
a
d
i
c
f
