Colloquium publications.

LINEAR SYSTEMS OF CURVES.

11

If the lines of two planes are related in the mode above described
to the curves of a system, the planes are thereby projectively
related to each other.
As to the second matter, it is needful to show that the elements
used as auxiliaries in Lemma 1 have unique analogues in a system, triply infinite, of curves conforming to the second definition.
What were called points there have become curves here, hence
the lines and planes must be replaced by ool and oo systems
of curves. We need only examine, accordingly, whether the
postulate: a line and a plane intersect in one point, retains its
validity. Let a " line" be given by two curves, a "plane" by
three; or to adhere more closely to the definition, consider an S,
given by two points, a and b, and an,2 consisting of all the
curves of the 003 system S3 that pass through a third point c.
Then will Sk and S2 have in common one and only one curve.
For in the S3 there is an S2 containing the point a; in this S2
there is one curve C that contains the points b and c (and by the
explanations of the above theorem we see that it must contain all
the intersections of any two curves fixing the S). As containing
a and b it lies in S; as containing c it lies in S,, and as containing these three arbitrary points it is by the definition unique.
Therefore, all the constructions of Lemma 1 have their unique
analogues in the system S3.
We conclude that the transition from an oo2 system to one 003
is possible, and that for r = 3 the first and second definitions
are equivalent. Mutatis mutandis, the induction from r = m to
r = mq + 1 can be made by similar means. Recapitulating we
have therefore the theorem:
An oor algebraic system of irreducible algebraic curves upon any
algebraic surface is linear if either (1) its elements can be put in a
one-to-one correspondence, projectively, with the hyperplanes of an
r-fold space; or (2) if through r generic points of the surface there
passes one and only one curve of the system. For r > I these two
defining properties can be inferred, each from the other.