Colloquium publications.

126

THE MADISON COLLOQUIUM.

for which D is the hypersphere, but includes at least all functions
admitting a properly discontinuous group of type (1) and meromorphic in a domain D defined by a relation
g(u', a", v', v") < 0,
where g is a quadratic polynomial. Moreover, the functions
cannot be continued analytically beyond D.
In this same year Picard* began the investigation of functions
which admit a group of transformations of type (2). These
functions he denoted as hyperabelian functions, since the first
problem which he was led to study concerning them was one
related to the Abelian thetas and the Abelian modular functions,
p = 2. The classes discussed yielded functions with properties
analogous to those of the hyperfuchsian functions.
Generalizations. In a systematic development of the theory
of the automorphic functions of several complex variables a
question of first importance is that of the existence of a fundamental domain belonging to a properly discontinuous group.
A solution of this problem for such groups of projective transformations in n variables, - groups of type (1), - has been given
by Hurwitz.t
The extension of the two theorems of Weierstrass, ~ 4, for
the case of automorphic functions in n variables has been treated
by Wirtingert by the aid of methods of the general theory of
functions.
A systematic generalization of the theory of a class of hyperabelian functions was outlined by Hilbert and elaborated by
* Notes in the Comptes Rendus for 1884, followed by a systematic presentation in Journ. de Math. (4), 1 (1885), p. 87. Cf. further Bourget, Toulouse
Ann., 12 (1898), p. D 1; Humbert, Journ. de Math. (5), 5, 6, 7, 9, 10 (1899 -1904), and (6), 2 (1906).
t Math. Ann., 61 (1905), p. 325.
tSitzungsber. der Wiener Akad., 108 (1899), p. 1239. For the special case
of hyperabelian functions of n variables cf. Blumenthal, Math. Ann., 56
(1903), p. 510; ibid., 58 (1904), p. 497. Picard had long since used the second
theorem, stated for automorphic functions of two variables; cf. Journ. de
Math. (4), 1 (1885), p. 313.