Colloquium publications.

18

THE BOSTON COLLOQUIUM.

nuovo,* who discussed under only one specializing restriction the
surfaces whose plane sections are of deficiency 3. These are of
four kinds, so far as numerated, and not all rational.t In looking
for other possible extensions, it should be remembered that there
are other classes of highly specialized curves, differing from the
hyperelliptic in the degree of the singular series of special groups
which occurs upon them. Of such classes, individual curves have
received some study, but linear systems little or none.
CHAPTER 4.
Linear Exact Differentials of the First Kind on an Algebraic
Surface.
~ 1. The Existence of Integrals on Given Surfaces.
WHEN the theory of integrals upon algebraic curves was extended to surfaces, the first step was the discussion of double
integrals. These have been described already (Chapter 1), and
attention has been called to two important numbers, characteristic
of a surface, to which they give rise, the geometrical and the numerical deficiency. Every surface above the lowest orders possesses
double integrals of the first kind, everywhere finite, unless its
singularities have become too numerous. The increase of singular
points and lines causes a diminution of the geometrical deficiency,
pg. Double integrals and their classification were introduced by
Clebsch and Noether about 1870. Fifteen years later a different
and even more interesting extension of curve theory to surfaces
was made by Picard.+ The new integrals that he introduced
are simple integrals whose path of integration is restricted to
*" Sulle superficie algebriche le cui sezioni sono curve di genere 3." Torino
Atti, vol. 25 (1890).
t If the surface is of order above the fourth, with plane sections all of deficiency
3, it is rational. See Castelnuovo and Enriques "Sopra alcune questioni fondamentali nella teoria delle superficie algebriche," Annali di Matematica, ser.
3, vol. 6 (1901), esp. Sec. V, 16.
t "Sur les integrales de diffirentielles totales algebriques de premiere espece,"
Jour. de math., ser. 4, vol. 1 (1885).