Colloquium publications.

FUNCTIONS OF SEVERAL COMPLEX VARIABLES.

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admit two-dimensional cycles, and it is these that form the
analogue of the linear cycles in the case of the algebraic functions
of a single variable. With these are connected the double
integrals of Noether.
The methods employed in these early geometric investigations
are largely those of intuition and analogy. Picard recognizes
this fact, but points out that his chief object was to throw light
on a theory at that time wholly new.
Geometry on Algebraic Curves and Surfaces. The purely algebraic theory of the geometry of systems of points on algebraic
curves has been extended to algebraic surfaces and systems of
curves lying on them.*
The Point of View of the Theory of Numbers. The methods of
the theory of algebraic numbers, first extended to the algebraic
functions of a single variable, have been used by Henselt for
the study of algebraic functions of two variables. In his treatment of the theory of the algebraic functions of a single variable
Weierstrass had used purely algebraic methods. Hensel describes his own methods for algebraic functions of two variables
as the direct generalization of Weierstrass's methods.
In a preliminary study of these functions Hensel deduces
series developments which apply to the neighborhood of a
branch-line or of a multiple-line of the surface. The form of the
development in the neighborhood of a finite point, which we will
take as the origin (0, 0, 0), is the following:
z = e,(xX)(y - y0)llb + e2(x)(y - yo)2b +..
where b is a positive integer. The coefficients ek(x) and the
variable yo are analytic functions of i, where
* t~~~~~ = x1l/a
and a is a positive integer. In fact, the equation of the branch
* Noether, Math. Ann., 2 (1870), p. 293; ibid., 3 (1871), pp. 161, 547; ibid.,
8 (1875), p. 495. Picard et Simart, Fonctions algebriques de deux variables,
vol. 2.
t Adcta, 23 (1900), p. 339; Jahresber. D. M.-V., 8 (1899), p. 221.