For a regular closed curve on the plane it is known that
E
=
I
+
X
+
1
2
F
E = I + X + \tfrac {1}{2}F
where E, I, X and F are the numbers of external double tangents, internal double tangents, self-intersections, and inflexion points respectively. It is proven here that if
F
=
0
F = 0
then I is even and
I
⩽
(
2
X
+
1
)
(
X
−
1
)
I \leqslant (2X + 1)(X - 1)
. Furthermore, examples are given which show that if the four tuplet (E, I, X, F) of nonnegative integers satisfies (a) F even, (b)
E
=
I
+
X
+
1
2
F
E = I + X + \tfrac {1}{2}F
, and (c) if
F
=
0
F = 0
then I is even and
I
⩽
X
(
X
−
1
)
I \leqslant X(X - 1)
, then there is a regular closed plane curve which realizes these values.