Colloquium publications.

FUNCTIONS OF SEVERAL COMPLEX VARIABLES.

141

the irreducible equation f(x, y, z) = 0, it is possible to associate
with this function the surface in projective space given by setting
Xi X2 X3
x= —, y=-, zxo xo xo
It is, however, also possible to put
Xi X2 Zo
and still again to set
X Yl Zl
x=- -- y=-, =-.
XO yo Zo
There is another geometry that is well known,- the geometry
of reciprocal radii, or the geometry of inversion. It would, of
course, be a proceeding entirely coordinate with that which has
been set forth above to extend the finite space of n complex
variables to the space of that geometry.
These questions could not arise in the case of analytic functions
of a single complex variable, for there the infinite region of projective geometry, the geometry of inversion, and the space of
analysis are the same, namely, one point.
For the case of two complex variables, the infinite region of
the space of analysis and the infinite region of projective geometry
are different, and moreover the space of analysis and projective
space can no longer be transformed on each other in a one-to-one
manner and continuously. But the space of analysis is transformable in a one-to-one (but non-real) manner, and continuously,
on the space of the geometry of inversion. When the number of
complex variables exceeds two, all three spaces are distinct.*
* For a detailed treatment of these questions cf. a paper by Bocher,
Bulletin Amer. Math. Soc. (2), 20 (1914), p. 185. We note that the infinite
region of the space of analysis consists of n complex (n-l)-dimensional
manifolds (hyperplanes) which have as their sole common point the point
(oo, 0o,..., oc).