Colloquium publications.

210

THE MADISON COLLOQUIUM.

The second method,* however, yields the desired results with
essentially the same machinery as that used for the n-leaved
surface, but with no excepted cases whatever.
The final result in either case is that the formulas for which
(11) and (12) are representative hold unchanged for the t-plane.
Thus, for example, if t, r, t', i' be any four distinct points of, we shall havet
(14) I~t, log (t - t')(r- r')
(14) t = (tlog ) -') + W(t, T; t', X),
where W(t, r; t', r') is analytic in its four arguments throughout
the cylindrical region (~, A, A, 5) inclusive of all boundary
points.
The Transformations of the Functions. Let
(15) ta = La(t), = 1,..,p,
be the linear transformation of G which carries Aa into Aa+;
and let
(16) tp+a = Lp+a(t), a = 1, *, p,
carry Ba- into Ba+. The inverse,
t' = L-l(t),
will carry A + into A-; and similarly for (16).
The transformations (15), (16) together with their inverses
constitute a system of generators for the automorphic group G.
Any single-valued function on F which has no other singularities
than poles goes over into a single-valued function of t having no
other singularities than poles within the unit circle I t I < 1 and
(with the single exception of a constant) having the circumference
of the circle as a natural boundary.
The Abelian integrals of the first and second kind also go over
into single-valued functions of t having no other singularities
* Cf. Fricke-Klein, Automorphe Funktionen, vol. 2, chap. 1.
t The notation for the dependent variable here and in the following is the
same whether the arguments are taken in the t-circle or on the z-surface. It
will be clear from the context each time which is meant.