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.

160

The implicit algebraic function.

Bk. II. oh. II. V.

precisely into this value v' and into none of the others. For, at each
point through which z passes, there is always only o n e value that differs
from,w by less than ~D. If then w were to change continuously to a
value differing from w1 by more than I-D, this difference should somewhere
become equal.to I D; but this cannot be for any point in the circle. Thus
too since t' lies in the circle round z', the difference between v' and
w' is smaller than -1D. Similarly, to t" belongs one value v", differing
from the value w' by less than -D, since t" is on the circumference
of the circle drawn round z'. This value is obtained by travelling
along the curve (2) from t' to t"; for were w to assume a different
final value, it would be one differing from w' by more than ~D.
Therefore there should be upon the path t't" a point at which the
difference is -I-D; and this again is excluded, because t' and t" are
within and upon the circle round S'. Since t" lies in the circle round
the centre z" we have:
mod [v"- w"] < -ID.
In like manner it can be seen that for the values at t" and zv:
mod [VV - w"] <- 4D,
and from these points we arrive at the values W and V. Here the
inequalities are:
mod [ WY- tw] < i-D, mod [ - wu, < - D.
Hence follows that mod [W - V] < D.
Now since by hypothesis the n different values of w at the point
Z differ from each other at least by D, W and V cannot signify two
different values, so that we must have V== W. Therefore the curve (I)
and the curve (2) lead to the same final value. From curve (2) we can
pass over to a curve (3) closer to (II) and proceeding thus we must
ultimately be able to arrive at curve (II). For, all the radii h are
of finite assignable magnitude therefore it is not possible that the
number of steps can be infinite. Such a progressus in infinitum can
only occur when the interpolated curves are approaching a critical
point of w: for, in the immediate neighbourhood of such a point no
circle can be determined within which always one only of the
corresponding roots differs from wu by less than -I D: the quantity
indicated by D), here converges to zero. If in the included domain
there be a non-essential singular point that is not also a critical point,
the theorem still holds. For, although the algebraic function becomes
infinite in this point, yet it retains the character of a rational function
and remains unique. In fact if we surround the point by a circle of
arbitrarily small radius, then, while z goes round this circle, that root,
whose amount increases beyond all limits as the singular point is
approached, passes through a continuous series of values, that returns