The present memoir is intended to be supplementary to that "On the Double Tangents of a Plane Curve.” I take the opportunity of correcting an error which I have there allen into, and which is rather a misleading one, viz., the emanants U
1
, U
2
,.. were numerically determined in such manner as to became equal to U on putting (
x
1
,
y
1
,
z
1
) equal to (
x, y, z
); the numerical determination should have been (and in the latter part of the memoir is assumed to be) such as to render H
1
, H
2
, &c. equal to H, on making the substitution in question; that is, in the place of the formulæ U
1
= 1/
n
(
x
1
∂
x
+
y
1
∂
y
+
z
1
∂
z
)
2
U, U
2
= 1/
n
(
n
-1)(
x
1
∂
x
+
y
1
∂
y
+
z
1
∂
z
)
2
U, &c., here ought to have been U
1
= 1/(
n
- 2)(
x
1
∂
x
+
y
1
∂
y
+
z
1
∂
z
)U, U
2
= 1/(
n
- 2)(
n
- 3) (
x
1
∂
x
+
y
1
∂
y
+
z
1
∂
z
)
2
U, &c. The points of contact of the double tangents of the curve of the fourth order or quartic U = 0, are given as the intersections of the curve with a curve of the fourteenth order II = 0; the last-mentioned curve is not absolutely determinate, since instead of II = 0, we may, it Is clear, write II + MU = 0, where M is an arbitrary function of the seventh order. I have in the memoir spoken of Hesse’s original form (say II
1
= 0) of the curve of the fourteenth order obtained by him in 1850, and of his transformed form (say II
2
= 0) obtained in 1856. The method in the memoir itself (Mr. Salmon’s method) gives, in the case in question of a quartic curve, a third form, say II
3
= 0. It appears by his paper “On the Determination of the Points of Contact of Double Tangents to an Algebraic Curve,” that Mr. Salmon has verified by algebraic transformations the equivalence of the last-mentioned form with those of Hesse; but the process is not given. The object of the present memoir is to demonstrate the equivalence in question, viz. that of the equation II
3
= 0 with the one or other of the equations II
1
= 0, II
2
= 0, in virtue of the equation U = 0. The transformation depends, 1st, on a theorem used by Hesse for the deduction of his second form II
2
= 0 from the original form II
1
= 0, which theorem is given in his paper “Transformation der Gleichung der Curven 14ten Grades welche eine gegebene Curve 4ten Grades in den Berührungspuncten ihrer Doppeltangenten schneiden,”
Crelle
, t. lii. pp. 97-103 (1856), containing the transformation in question; I prove this theorem in a different and (as it appears to me) more simple maimer; 2nd, on a theorem relating to a cubic curve proved incidentally in my memoir“ On the Conic of Five-pointic Contactat any point of a Plane Curve”, the cubic curve being in the present case any first emanant of the given quartic curve: the demonstration occupies only a single paragraph, and it is here reproduced; and I reproduce also Hesse’s demonstration of the equivalence of the two forms II
1
= 0 and II
2
= 0.
Some Properties of the Exeter Transformation
