Colloquium publications.

168

THE MADISON COLLOQUIUM.

single piece. Finally, the points of (c) yield one or more 2-dimensional pieces of the 3-dimensional boundary of (B, B') just
referred to, and they also lie in the points of (b).
A further aid toward a geometric realization of the hypotheses
is obtained if we picture the cylindrical region (B, B') as a rectangle in the plane of analytic geometry. Here, as in the use of
that plane in the study of plane curves when the complex points
are admitted to the discussion, we have, it is true, only a twodimensional figure for a four-dimensional set of geometric objects;
and we have to work by analogy.
-— ______________.____ ___ _ _ *
FIG. 2.
Condition (a) is now seen to refer to the points of a narrow
strip that courses the large rectangle Imno, the latter representing
the region (B, B'). Conditions (b) and (c) have to do merely
with points of the boundary, which lie in the sides lo and mn.
In the conclusion, the function is extended over an enlarged
region dimensionally coordinate with the slender strip of condition (a).
The extension of the theorem in the above formulation to the
case of n-variables is obvious.
For three variables, the geometric interpretation last considered
leads to a rectangular parallelepiped, coursed by a slender one
with parallel faces, and the further conditions of the theorem
are interpretable in terms of regions and curves lying in the faces
of the large parallelepiped.
Another form of the hypotheses of the theorem, somewhat
less general, but more compact, consists in requiring the function
f(l * ' *, Xn) to be analytic