Colloquium publications.

FUNCTIONS OF SEVERAL COMPLEX VARIABLES.

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having the same properties. Let r be an arbitrary point of A,
and let it be held fast. Form the function
T (t, T)
Q(t, r) ~
Then this function, regarded as a function of t, will be analytic
in K except for removable singularities in the points t = r and
in the images of this point under the group G. Let it be defined
in these points as equal to its limit. The new function is analytic
without exception in K, and does not vanish there.
From (d) it follows further that this function is invariant of
the transformations of G. It is, therefore, a constant, as can be
seen at once by transforming it to the n-leaved surface F. Hence,I'(t, T)
(t,r ) _= f(T), (t, T) = f(T) Q(t, r), f(r) + 0.
This last relation is an identity in t, r, and hence can equally
well be written in the form
T(r, t) = f(t) Q(r, t).
Now apply the property represented by (b). It follows that
-(t, r) = -f(t) (t, ).
Hence
f(t) = f (r),
and this completes the proof.
From the foregoing result it is seen that the properties (a),
*.., (d) can serve as the basis for an independent definition
of the prime function Q(t, r). Thus the function might be
represented by an infinite product, as Weierstrass defined his
elliptic a-function. And just as Weierstrass made the latter
the basal function for the whole theory of the elliptic functions,
so the algebraic functions of deficiency p > 1, and their integrals,
can be represented in terms of Q(t, r). We proceed to give the
fundamental formulas.