Colloquium publications.

190

THE MADISON COLLOQUIUM.

the transformed configuration F1 = 0 the same meaning that
i ', * * have for F = 0, i. e., Si, "*, e are invariant of
the transformation. And now, if the neighborhood of Pi is
suitably restricted, the number of the ordinates u corresponding,
for a given (x, * * * ), to the ith region i', i = 1, * *,,
will reduce to unity. Thus the new 1 will equal f, and to each of
the new points (xi, * *, Xn-i, Xni)) will correspond but a single u.
It thus appears, that, in general, on the (2n - 2)-dimensional
analytic manifold or manifolds defined by the equation A = 0
the multiple roots of F are single-valued and analytic except
along certain (2n - 4)-dimensional analytic manifolds.
We can now proceed to these latter manifolds and prove a
similar theorem for them; and so on.
The reasoning here used is akin to that employed in the proof
of Weierstrass's theorem, ~ 8.
~ 5. SINGLE-VALUED FUNCTIONS ON AN ALGEBROID
CONFIGURATION
Let U be uniquely defined in the ordinary points of the algebroid configuration (1), i. e., the points in which u is not a multiple
root of (1), and let it be analytic in such points. If, furthermore,
U remains finite, then, in the above points,
G(u, xl, '", xn)
F'(u, xi, *, n)'
where G is analytic in the point (0, 0, * *.., 0) and vanishes when
F' vanishes on the manifold. Moreover, U is an algebroid function of (x1, * *.., x) in the neighborhood of the origin.
It would be a mistake, however, to think that when U satisfies
the above hypothesis, the limiting values of U in the singular
manifold behave as did the coordinates, u, xn, etc., in the cases
discussed in ~ 4. The following example will show what may
arise. Let
F- u2- x(y2 - z2)(y2 - k22) = 0,
where k is real and 0 < k < 1, and the independent variables