Colloquium publications.

116

THE MADISON COLLOQUIUM.

It has the properties:
o(u + ti) = (u),
6(u + a) = e-2u-af(u);
and it has, moreover, a single root of the first order in the parallelogram F, two sides of which are the vectors (0, ri) and
(0, a).
By means of this function, doubly periodic functions can be
formed as follows. Let a a,, * *, ~ 1 *..,,* * be any 2n points
so chosen that
n n
a ak = ik)
k=1 k=1
and that, furthermore, the points of the parallelogram F that
are congruent to them are distinct. Then the quotient
M(u + ai) - * (u + an),(u + I1) '* * * (u + An)
will evidently represent a doubly periodic function with the
periods -ri and a.
The fundamental theta function of p arguments is given by
the following series:
p
r+2 ~ nkukM
zt(u1, up) = C Ee k' C * o,
where
1p
r = r(ni, * *-, un) = E akinknt, aki = alk,
k, 1=1
akl = rkl + iSkl
and the real part of r(xi, * *, Xp), where xi, *., xp denote real
variables, namely
p
ZE rklXkXl,
k, 1=1
is a definite negative quadratic form.
The function has the following properties, readily deducible