Colloquium publications.

122

THE MADISON COLLOQUIUM.

tions of several complex variables. He carries his proof through
only to the point of showing that the given function can be written
as the quotient of two Jacobian functions. The latter functions
are defined as follows.
Jacobian Functions. Let w, a = 1, *.., p; = 1,., 2p,
be a primitive scheme of periods, and let f(zi, *.., z,) be an
integral function of its p arguments. If, for every period from
this scheme, a relation of the form holds:
f(z + * * *,,z + qc- ) = eLP( )f(zi,... ),
where L3(z) is a linear (homogeneous or non-homogeneous, but
integral) function of zl, * * *, p, thenf is called a Jacobian function.
The Jacobian functions have been studied at length in two
memoirs by Frobenius,* and in a paper by Wirtinger. t A
Jacobian function can be expressed in terms of theta functions
of p arguments.
~ 5. AUTOMORPHIC FUNCTIONS OF SEVERAL VARIABLES
The brilliant results obtained by Klein and Poincare in the
early eighties, in their researches relating to the automorphic
functions of a single complex variable turned the attention of
mathematicians towards functions of several complex variables
which admit a discrete group of linear transformations into
themselves, and we find from that time to the present day a
steady stream of papers in this field.
Here, however, at the very threshold of the subject, two types
of groups present themselves, corresponding on the one hand
formally to the linear transformations of projective space:
(1) ( a'x + by+ c' a"x + b"y + c"
x y ax+ by+ c ax + by c )and on the other, to those of the space of analysis:
* Journ. fur Math., 97 (1884), p. 16 and p. 188.
t Monatshefte fur Math. u. Phys., 7 (1896), p. 1.