Colloquium publications.

208

THE MADISON COLLOQUIUM.

while the fourth is allowed to range over its circle, it appears
from (6) and (7) that IIHz is analytic in this variable alone.
Furthermore, from the considerations which have just preceded,
it is seen that when all four variables range over their circles,
II|~ remains finite in T. We infer, then, from the theorem of
II, ~ 5, in its restricted form that II is analytic in all four variables
regarded as simultaneous.
Next, let us consider the function II when zo =, the point z
lying in the circle about 0o. The points 0o, 7r0, wo are distinct
ordinary points. But it is necessary now to demand that z shall
not coincide with I. If we write
(I1) no; = log (z - ~) + (z, w, 77),
then t is defined at all points of T except those of the locus
z = i, and W is finite. It follows here, as in the earlier case, that
a[ is analytic in those points of T in which it is defined. And
now comes a typical application of the theorem of III, ~ 4,
relating to removable singularities. From it we infer that a
approaches a limit in each of the excepted points, and that, if
S is defined there as equal to its limit, then a2 will be analytic
there.
Similar formulas hold for other coincidences of the points
z0,, o, o0, o. Thus, when all four points coincide,
(12) IIn = log (z- ~)(w_ + A(z, w, i, v),
where A is analytic in all four arguments, regarded as simultaneous, in the point in question.
~ 5. THE FUNCTIONS IN THE AUTOMORPHIC FUNDAMENTAL
DOMAIN
We proceed now to transfer all the functions from the n-leaved
Riemann's surface of the z-plane to a fundamental domain W
in the unit circle of the t-plane. The relation between z and t
shall be expressed by the equation

(13)

Z = (t), or t = (z).