Colloquium publications.

FUNCTIONS OF SEVERAL COMPLEX VARIABLES.

225

~ 12. THE ALGEBRAIC FUNCTIONS
A single-valued function on F having no other singularities
than poles yields an absolute invariant of the group G, the new
function, considered in K, being analytic except for poles.
The necessary and sufficient condition which 2n points of A,
ak and Tk, k = 1, **', n, must satisfy if they are to be the zeros
and poles of such a function F(t), is found by contour integration* to be the following:
(i) W,(7al)+ * *. +Wu(arn) —'(7Tl)+ * '* +Wa(r7,) (mod. periods),
i. e.,
n p
(i') ZwaT = Ma 7i + Vj aj,
k=l j=l1
where a = 1, *., p and the Ju, vj are integers.
If one of the above points, as Tn, be replaced by an equivalent
point under the group G, the coefficients /, vj will thereby be
modified. It is evidently possible to choose the latter point so
that these coefficients will all be zero, and we suppose this done.t
(i") Z 'kk = 0
k-=1
where, now, (1,..., un and 71,..., rTn- lie in A, while Tn may
not.
More generally, the points,,.., n and 1,, Tn may be
any 2n points satisfying (i") and such that the points of:
equivalent under G to ai, * *, A, are all distinct from the points
of a equivalent to r1,.*, Tn.
To express the function F(t), or to prove the existence of such
a function, the procedure is similar to that in the elliptic case,
where a(t - r) corresponds to the present Q(t, r). Form the
* The integrals which are to be extended round the boundary of a are
- f da log F(t), - f wa(t) d log F(t).
t It is sufficient for the application which follows that merely the vi be
reduced to zero.