1.3. DEGREE OF Á DUAL SURFACE (SALMON) 3
is on 5QX Ð SQ2 if and only if the tangent plane to S at Ñ eontains l. Thus for S
nonsingular of degree n, and Qi, Q2 general,
deg(5v) = # ( 5 Ð SQl Ð SQ2) = n(n - l) 2 .
As before, all singular points of S are contained in 5 Ð SQA Ð SQ2 . If Ñ is an
isolated singular point of S, its eontribution to the total n(n — l) 2 is the intersection
For example, the eontribution of an ordinary double
point is two, so
= n(n — l)
— 26 if S has ä ordinary double points.
If S is singular along a curve C, however, a new phenomenon oecurs, a Prob-
lem of excess intersection'. how to compute the eontribution of C to the total
intersection n(n — l)
, so that n(n — l)
diminished by this eontribution, and by
contributions of other singular points, yields
Salmon initiates a study of
the eontribution of a curve C to the intersection of three surfaces in space when
C is a component of their intersection. For example, if C is a line, he gives its
eontribution as ra + n + p — 2, where m, ç , ñ are the degrees of the surfaces. Salmon
justifies this by saying that the answer must be independent of the choice of surfaces
of given degrees, and then he calculates it directly in the degenerate case when the
flrst is the union of a plane containing C and a general surface of degree m — 1.
This surface meets the other two surfaces in (m — l)np points, m — 1 of which are
on the line C. The plane meets the other two surfaces in curves of degrees ç — 1
and ñ — 1 in addition to C\ these curves meet in (n — l)(p — 1) points. The total
number of points of intersection outside C is therefore
(m — l)np— (m — 1) -f (n— l)(p— 1) = mnp— (m + n + p — 2),
as asserted. In case C is a double line on the first surface, he calculates its eontri-
bution as m + 2n + 2p — 4 by working out the case where this surface is the union
of two surfaces containing C.
If C is a double line on a surface 5 of degree n, this analysis predicts 5n — 8
as the eontribution of C to the intersection of S with SQX and SQ2. However, as
Salmon points out, there are special points on C, called pinch points (or "cuspidal"
points), where the two tangents planes to S eoineide.