Colloquium publications.

150

THE MADISON COLLOQUIUM.

From this theorem it follows that, when the function Sp(r)
exists at all, its interval of definition reaches back to the origin:
O < r < R.
If the function So(r) exists and is constant in a portion of the
interval of definition, then 9p(r) is constant clear back to the
beginning of the interval.
The function o(x) possesses a finite forward derivative and a
finite backward derivative, neither of which is positive. The
same is true of the function s = p(r).
If xi and x2 are any two points of the interval of definition of
@(X):
-oo < i < x2 < log R,
neither of the above-named derivatives in the point x2 exceeds
either one of the derivatives in x1.
From these results it is clear that, if P is any point of the
curve (5), then a straight line whose slope is negative or nil can be
drawn through P, such that the curve nowhere rises above the line.
By means of such lines, — " tangents," as Hartogs calls them,
-Hartogs and Faber* show that the fundamental property (2)
is the only condition which the function Sp(r) must fulfil. In
other words: Let p (r) be any function of r which is defined throughout an arbitrary interval 0 < r < R, is positive there, and is
subject to the condition (2). Then there exists a double power
series
"Cm,n X yn
to which the numbers r, s correspond as associated radii of
convergence, where
s = o(r), 0 < r < R.
~ 8. HARTOGS'S FUNCTION RX
In a number of his investigations Hartogs makes extended
use of a function R, which can be defined as follows.t Let
* Hartogs, Math. Ann., 62 (1905), p. 84. Faber, 1. c. Since Faber does not
introduce the logarithm, the tangents appear as his W-curves.
t Hartogs, Dissertation, and Math. Ann., 62 (1905), pp. 24, 25. The
notation there used is Rx0.