Colloquium publications.

THE BOSTON COLLOQUIUM.

This can be none other than C3, hence the curve which is determined by A, and the generic point P will contain also A2. By
parity of reasoning it must contain as well A3, *... *, A. But as P
was any point, C3 was any curve through A1, consequently every
curve of (C) that contains an arbitrary point At must contain also
n - 1 other determinate points, as asserted by the lemma.
The principal theorem can now be proven if two facts are established. First the theorem should be found to subsist for the
particular case r = 2, so that the base may be provided for a
mathematical induction. Then, secondly, the mode of induction
employed in Lemma 1 must be shown to be applicable to a system of curves conforming to the second definition.
PARTICULAR THEORE-M. A doubly infinite algebraic systeI of
irreducible algebraic curves upon any algebraic surface can be
brought into a one-to-one relation with the system of all lines in a
plane by assigning to four arbitrarily chosen curves of the system
(no three through one point), four arbitrarily chosen lines of the
plane (no three through one point), as corresponding lines, and by
requiring further that to curves having a common point shall correspond lines with a point in common.
To prove this, associate every set of vm points, such as the
Al, A2,..., A, of Lemma 2, together as one element A. Then
there is upon the surface an oo2 system of C's and a second system
of A's related thus: Two generic C's determine one A and two
A's determine one 0. Now these are precisely the incidence relations upon which depends the familiar proof that four lines of one
plane and four of another determine a projectivity of the two systems of lines; here the lines and points of the one plane are
replaced by elements C and A. The requisite of continuity is
provided for by the hypothesis that the system is of algebraic
character. Therefore the lines of a plane and the curves of the
system (C) stand in a one to one relation, as asserted by the
theorem. This relation is called projective, meaning that it is
independent of the particular four pairs, line and curve, that may
be selected to determine the correspondence. Otherwise stated: