The author remarks that his paper was intended as supplementary to Mr. Cayley’s Memoir “On Curves of the Third Order” (Philosophical Transactions, 1857, p. 415). He establishes in the place of Mr. Cayley’s equation, p. 442, a fundamental identical equation, which is as follows, viz. if substituting in the cubic U,
x
+ λ
x'
,
y
+ λ
y'
,
z
+ λ
z'
for
x, y, z
, the result is U + 3λS + 3λ
2
P + λ
3
U'; so that S and P are the polar conic and polar line of (
x', y', z'
), with respect to the cubic, viz. 3S =
x'
d
U/
dx
+
y'
d
U/
dy
+
z'
d
U/
dz
; 3P =
x
d
U'/
dx'
+
y
d
U'/
dy'
+
z
d
U'/
dz'
; and if making the same substitution in the Hessian H, the result is H + 3λΣ + 3λ
2
Π + λ
3
H', so that Σ and Π are the polar conic and polar line of the Hessian—then the identical equation in question is 3(SΠ — ΣP) = H'U - HU'.