Colloquium publications.

LINEAR SYSTEMS OF CURVES.

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when taken upon the surface between any two arbitrary points.
If the surface f= 0 is a cone, such differentials exist, for they
are the abelian differentials of first kind upon its plane sections.
Picard proves* that if the surface f = 0 have no multiple points
or curves, then no such differential can exist upon it. There are
however surfaces of all orders above the third which contain (or
admit) one such integral; others, from the sixth order upward,
which admit two, and so on. These surfaces and the mode of discovering them and of defining them have been the occasion of
some of the most interesting studies of Picard and Humibert. The
elementary part of Picard's first paper upon this topic I shall give
in some detail, indicating in conclusion certain points that might
prove worthy of further study.
CHAPTER 2.
Linear Systems of Curves on an Algebraic Surface. The Two
Geometric Definitions are Concordant.
IN plane geometry a linear system of algebraic curves is defined
analytically by an equation containing linearly and homogeneously
two or more parameters; as for example:
Xoo + X\10 + 202 +... +XK K= o,
the X's being parameters, and the O's a set of polynomials homogeneous of like degree in the current coordinates. This is called
a K-fold infinite (ooK) linear system. As we restrict our field to
include only systems defined by fixed base points, the curves
oi! = 0 must be supposed all to contain the base points of the
system. In a plane such a system may be studied by means of
its equation, but for other surfaces one must either assume an
analytic representation as definition, or else take such geometric
features of a plane linear system as seem most important and
transfer them to sets of curves on surfaces in general. \We follow
* Picard et Simart: Theorie des fonctions aclqbriques de deux v riables independantes, vol. 1 (1897), pp. 119-120.