Colloquium publications.

130

THE MADISON COLLOQUIUM.

Such an integral is a function of two independent variables,
and these may be taken as x, y or y, z or z, x.
A division of such integrals into three classes, corresponding
to the three classes of Abelian integrals, at once suggests itself.
In the first paper above referred to Picard studies the integrals
of the first class, namely, the everywhere finite integrals.* He
finds here a situation diametrically opposite to that in the case
of the Abelian integrals. If f(x, y) = 0 is an irreducible algebraic equation of degree greater than 2, there will in general
exist integrals of the first class corresponding to it; it is only
when the curve is highly specialized (unicursal) that this is not
the case.
To the non-specialized algebraic surface of arbitrary degree,
however, there correspond no integrals of the first kind with the
trivial exception of a constant. A special class of surfaces and
integrals is treated, the former being those which can be uniformized by means of quadruply periodic functions of two independent variables.
It was in these papers that Picard began the study of questions
relating to the connectivity of the surfaces which present themselves. The points of an algebraic surface fill a four-dimensional
region, -be that region assumed as a four-dimensional manifold
in space of six or more dimensions, or as a multiple-sheeted
Riemann manifold, or as a fundamental domain, for which the
parallelogram of periods is the prototype. In this fourdimensional manifold the linear cycles (closed curves) and the
two-dimensional cycles (closed surfaces) are of especial importance. Picard finds the striking result that, in the case of a
non-specialized algebraic surface, any linear cycle can be drawn
together continuously to a point. This fact explains, -or is
explained by, -the non-existence of integrals of the first class
on such a surface.
On the other hand, a non-specialized algebraic surface does
* Picard's first publication relating to the integrals of the second class
appeared in the Comptes Rendus, 100 (1885), p. 843.