Colloquium publications.

FUNCTIONS OF SEVERAL COMPLEX VARIABLES.

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z =;;, f(z = =(')
the function po(z'), which is not defined in the point z' = 0, but
is analytic in the rest of the neighborhood of this point, shall
have a removable singularity in the point z' = 0.
Returning to functions of several variables, let us raise again
the question, why introduce the space of analysis? A contribution toward an answer to this question is to be found in the
two theorems of ~ 4, below. For simplicity, let us restrict ourselves to the first one. This theorem is not true if our hypothesis
be merely that the function shall be meromorphic in every point
of finite space. Some further hypothesis relating to its behavior
at infinity, or to the behavior of the function when subjected
to certain transformations, is essential. And now Weierstrass
supplied this condition, - or appears to have done so, - in the
way indicated above.
But is this the only way in which this end can be attained
without doing violence to simplicity or custom? By no means,
as we shall presently see.
Projective Space and the Space of the Homogeneous Variables.The space most familiar to the geometers is projective space, and
this space is mapped in a (1, oo)-fold manner on the space of
n + 1 homogeneous variables x0, xi, * *, Xn. This latter space
is the whole finite space whose points are (xo0, xi,., xn), where
each coordinate ranges over its whole finite Gauss plane, the
one point (0, 0, * -, 0) being excluded. We will speak of it as
the space of the homogeneous coordinates.
The functions considered in this space had their origin in
projective space (itself but an amplification of an ordinary
finite space), and are homogeneous in the n + 1 variables,polynomials, algebraic functions, and such transcendental
functions as are suggested by the names of Aronhold, Clebsch and
Gordan, Klein, and their school.
Might it not have been possible to choose the complementary
hypothesis in Weierstrass's theorem is ~ 4, not with reference
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