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.

~ 98.

Examples.

167

In order to establish in what manner the roots interchange about the
points j, let us start from a point z at the arbitrarily small distance r
from the origin on the axis of abscissa; the corresponding roots are
real and quam proxime:
W = _ r2 20 r 0 j/3= -_ /'3 - ao
W1 -2- ) IV = ---
Two of them are positive and one negative, and all three remain real
as z moves along the axis of abscissa towards the point f1. For then,
since complex roots can occur only as conjugate pairs, and moreover
the real constituent and the imaginary vary continuously, a transition
to complex values can only occur in points wherein the real constituents
are equal and the imaginary constituents vanish, that is to say, in
branching points. In the point ~, the value of the two equal roots is
V = + I1/'= + 23;
the two positive values interchange; it is a branching point for the
leaves 1 and 2; let its section be chosen so as not to cut the segments
0p2, 013.
Let us now consider a point arbitrarily near the origin upon the
right line Op2; when we proceed to this point along the circle with
radius r without crossing the negative axis of ordinates, we have:
4ir7 iAt 4i2t 4in
m0 =3 I 3 r
W1 _ re3, 2~ = J3r = - re ---rie 3, 3 = /3 r"e
2i1
A point upon Op2 is expressed by z = e 3, the value of the double
root for the point fi2 is
~~~4i~~~t V s~~~~~~~ectio
w == 2e. If then we
substitute: 7\
2i i 4iz /
z = Pe 3, w = w'e 3
in our equation 2), it takes /
the form:
w'3 - 3ew' + - - O, + 0
from which it appears,),
that w' behaves along 2 a
Op/2 just as w along the m ~
radius 01,, thus the two
positive values with the
4i'o
factor e 3 viz. w1 andw3, Fig. O.
interchange; therefore i3
is a branching point for the first and third leaf; let its section be
drawn so as not to cut the segments 01,, 03p3.