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.

~ 94. 95.

Analytic property.

159

at the point Z; it is therefore conceivable, that w, may pass over
into these n different values at Z, according as S travels by different
paths from 20 to Z.
When two different paths leading from zo to Z bound a finite
surface and neither enclose nor pass through a critical point, the values
acquired by w at the terminal point Z are identical. If on the other
hand there be any such points within the surface, the values may
be different.
When there is no critical point within the surface, a finite
quantity D can be assigned that will be the minimum absolute difference
between the various values of w belonging to any single value s.
With 0o as centre and radius h a circle can be drawn, such that for
all points within it, one and only one of the values at any such
point differs from the value w1 by a quantity with smaller modulus
than D, while the other values corresponding to the point must
differ from w1 by more than - D; for w1, as was proved, is a unique
and continuous function along every
continuous curve, and there can be
_/ X\^ only one value differing from w1 by,
/, \ less than 1D at each point in this
circle, because if there were two
such values they would differ from
each other by less than D. Let this
I / ^\ a9/ /2 circle intersect the curve (I) in
the point 2' and let us call the
orresponding value
wv + Aw === w'.
1 "h/ \, / SA circle with radius h' can be
drawn round a' as centre with a
C Fig.\ Jsimilar property; let it cut the
curve (I) in the point '"; we have
Fig'. 9.
"= w' + Aw, (mod Aw < -1D).
Repeating this process a finite number of times we arrive at a point
a' with the value wv and from this reach the point Z with the value
W -= w -+ Aw, (mod Aw < -D).
Now if we draw between (I) and (II) a curve (2) from z0 to Z, near
enough to (I) to intersect the v circles in the points t', t"... t, the
portion 2ot' must be within the circle round zo; t't" within the circle
round ';... tVZ within the circle round zv. To t', then, belongs one
and only one value w ==-v' differing from w1 by a quantity whose
modulus is smaller than ~D. Now we must observe that if z move upon
the curve (2) from 20 to t', w passes continuously from the value wi