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.

~ 95. 96.

Uniqueness along certain definite paths.

161

back to its initial value. For in this case in the equation f(sm, wn) == 0
we have at the point S = a, q(p(a) =-O, while pl (a) is different from
zero. Therefore if we put z = a + A a and consider the equation:
wnAO + wun- lA] + * An = 0, writing Ak == (pk ( + A a),
we can choose A a so as to make A0 smaller than an arbitrarily small
number A, while mod [A,] certainly remains greater than some finite
number B. Consequently it is true, in conformity with the proof
~ 93, of the equation:
Ao + w'A1 + w'2A, +.. w'An = — 0,
that it has for each value of A a one and only one root, for which
mod w'= mod - < A, or, mod w > d
i. e. when z has gone round the arbitrarily small circle, the final value
of w can coincide with no other than the initial value.
From the theorem follows further: If we make the argument s
begin at a point zo and go round a closed finite curve not including
any critical point, the final value of w1 at z0 is the same as that with
which it began.
96. These Theorems do not yet enable us to picture to ourselves
a branch of the function. For, having calculated the value w1 at a
point zo, we can arrive at any other point by very different paths and
any two of them may include a critical point; thus it is still possible
that we may obtain at each point different values according to the path.
The perfectly unique exposition, already exemplified in the explicit
irrational function, is obtained here also by Riemann's method of
adopting for the representation of z, instead of a single plane, n planes
fastened together along their branching sections. But this requires that
we should investigate more closely the properties of the critical points.
Let the point = 0 be a regular point for all values of the
function w; i. e. let n simple finite roots of the equation
f(zm, Wtn ) 0
belong to it. Now conceive n different planes lying one on another;
to each of them coordinate in the point - 0 one of the values
wO,.... ~, indexing each plane by a number 1, 2,... n. Further, mark
in each plane all those points 3, which are critical points for any values
whatever of w, and from these points to the point infinity draw curves,
intersecting neither themselves nor one another; in all the n planes
the curves proceeding from a point Pk are conceived to be identical.
Let us further coordinate to each point of plane 1 that value of w
into which w1~ changes continuously when the argument z is made
travel along a path not crossing any of the curves starting from the
p points. In this manner a perfectly determinate value of w belongs
HARNACC, Calculus. 11