It may be convenient to mention at the outset that, in the paper “On the Theory of Skew Surfaces”, I pointed out that upon any skew surface of the order
n
there is a singular (or nodal) curve meeting each generating line in (
n
-2) points, and that the class of the circumscribed cone (or, what is the same thing, the class of the surface) is equal to the order
n
of the surface. In the paper “On a Class of Ruled Surfaces”, Dr. Salmon considered the surface generated by a line which meets three curves of the orders
m
,
n
,
p
respectively : such surface is there shown to be of the order =2
mnp
; and it is noticed that there are upon it a certain number of double right lines (nodal generators); to determine the number of these, it was necessary to consider the skew surface generated by a line meeting a given right line and a given curve of the order
m
twice; and the order of such surface is found to be =½
m
(
m
—1)+
h
, where
h
is the number of apparent double points of the curve. The theory is somewhat further developed in Dr. Salmon’s memoir “On the Degree of a Surface reciprocal to a given one”, where certain minor limits are given for the orders of the nodal curves on the skew surface generated by a line meeting a given right line and two curves of the orders
m
and
n
and respectively, and on that generated by a line meeting a given right line and a curve of the order
m
twice. And in the same memoir the author considers the skew surface generated by a line the equations whereof are (
a
, ..)(
t
, 1)
m
=0 (
a'
, ..)(
t
, 1)
n
=0, where
a
, ..
a'
, .. are any linear functions of the coordinates, and
t
is an arbitrary parameter. And the same theories are reproduced in the ‘Treatise on the Analytic Geometry of Three Dimensions’ §. I will also, though it is less closely connected with the subject of the present memoir, refer to a paper by M. Chasles, “Description des Courbes à double courbure de tous les ordres sur les surfaces réglées du troisiѐme et du quatriѐme ordre”||.
Hilbert–Mumford criterion for nodal curves
