An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 96. 97.

The n-leaved Riemann's surface.

163

the left side of the (-curve. Either, this leaf has then on the right
side of the p-curve the values wl, in which case the three leaves:
1) with the values: wu on the left, tw on the right,
2) with the values: w2 on the left, w3 on the right,
3) with the values: W3 on the left, w, on the right
form a cycle, and constitute a winding surface of the second order:
on completing a circle therein round the point i, keeping it on the left,
we pass from leaf 1 into 2, in a further circuit from 2 into 3, and in a
third from 3 into 1; if we travel in the opposite direction along the
circle we come from leaf 1 into 3, from 3 into 2 and from 2 into 1.
Or else, the third leaf has new values w4 on the right; there is then a
fourth leaf which either concludes the cycle or leads to new values wt
and thereby to an enlargement of the cycle.
Accordingly certain cycles of values belong to each branching
section /; and an n-leaved Riemann's surface is produced by fastening
together the different leaves belonging to each cycle along all sections.
Throughout any arbitrary curve drawn upon this surface, whether it
cross branching sections or not, the algebraic function w is completely
unique and continuous. It can become infinite only in the non-essential
singular points a.
97. The only further remark we shall here make is on the values
of w at the point infinity. In our distribution of values to each
leaf, the point infinity occurs as a many-valued point, i. e. at the
two sides of the /3-curve that is a branching section in leaf 1, the
function w takes for z = oo both the values which ex. gr. w1 and w2
assume for this value of the argument. Within any finite distance
however small from the point infinity, taking the illustration (~79) from
the sphere instead of a plane, or in numerical language, however
great may be the value of z, complete uniqueness still prevails.
The character of the point infinity, as a regular point or as a
branching point with determinate cycles in regard to the different
values of w, reveals itself, when starting from a leaf 1 we construct
a circle surrounding only the point infinity and no other branching
point, and consider how w changes value along this circle. By a circle
surrounding only the point infinity, the Transformation by Geometrical
Inversion plainly shows we have to conceive a circle round the origin,
whose radius can be arbitrarily great, but at any rate must be so
great that it shall include all finite branching points. When we
go quite round this circle keeping the finite surface on our right, this
signifies a circuit of the point infinity that is included on the left.
Accordingly we have the relation:
When the circuit of all finite branching points does not change a
value w, the point infinity is not a branching point for wl; and conversely.
11*