In my “Memoir on Curves of the Third Order, ”I had occasion to consider a derivative which may be termed the "tangential” of a cubic, viz. the tangent at the point (
x
,
y
,
z
) of the cubic curve (*)(
x
,
y
,
z
)
3
= 0 meets the curve in a point (
ξ
,
n
,
ζ
) which is the tangential of the first-mentioned point; and I showed that when the cubic is represented in the canonical form
x
3
+
y
3
+
z
3
+6
lxyz
= 0 , the coordinates of the tangential may be taken to be
x
(
y
3
—
z
3
) :
y
(
z
3
—
x
3
) :
z
(
x
3
—
y
3
). The method given for obtaining the tangential may be applied to the general form (
a
,
b
,
c
,
f
,
i
,
j
,
k
,
l
)(
x
,
y
,
z
)
3
: it seems desirable, in reference to the theory of cubic forms, to give the expression of the tangential for the general form; and this is what I propose to do, merely indicating the steps of the calculation, which was performed for me by Mr. Creedy. The cubic form is (
a
,
b
,
c
,
f
,
g
,
h
,
i
,
j
,
k
,
l
)(
x
,
y
,
z
)
3
, which means
ax
3
+
by
3
+
cz
3
+3
fy
2
z
+3
gz
2
x
+3
hx
2
y
+3
iyz
2
+3
jzx
2
+3
hxy
2
+6
lxyz
; and the expression for
ξ
is obtained from the equation
x
2
ξ
=(
b
,
f
,
i
,
c
)(
j
,
f
,
c
,
i
,
g
,
l
)(
x
,
y
,
z
)
2
,—(
h
,
b
,
i
,
f
,
l
,
k
)(
x
,
y
,
z
)
2
)
3
— (
a
,
b
,
c
,
f
,
g
,
h
,
i
,
j
,
k
,
l
)(
x
,
y
,
z
)
3
(C
x
+D), where the second line is in fact equal to zero, on account of the first factor, which vanishes. And C, D, denote respectively quadric and cubic functions of (
y
,
z
), which are to be determined so as to make the right-hand side divisible by
x
2
the resulting value of
ξ
may be modified by the adjunction of the evanescent term