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.

~ 199. The irreducible algebraic function defined by its irregular points. 375
s AZ(, O)
p (z))
where f is some function of the order m with respect to z, and
9(9() == (Z - Ca)' (z - C2jY... (z - c^e)'it ( -(- ),... (Z - /V)V
Accordingly
S.p (z) =f(, )
is an integer function of the mth degree in 2 and an integer function
of the nth degree in 6.
Hence, inasmuch as this polynomial f(zm, an) of the nth degree
in o vanishes whenever 6 assumes one of the values w1, we,... w,,;
w must satisfy, or, as the enunciation asserts, be root of the algebraic
equation:
f(i'", zV) == 0.
This algebraic expression is irreducible, i. e. it cannot be
resolved into rational factors of a lower degree in w, provided the n
generally different values of the function w are connected in such a
way that, by suitable choice of paths which enclose the branching
points, any value wi of the function can be carried over continuously
into any other wk; in other words: when the n-leaved surface requisite
for exhibiting the function w uniquely is connected not only in separate
points but along entire branching sections.
For, if f (zr, wn) break up into the product g (, w) ~ h (, w), since
each of these factors is of a lower degree than the nth in w, neither
of them can vanish for all the n values of w, they must therefore
both vanish for every value of z, ex. gr.
g (Z, Wi) = O, h (Z, 'WUk) = 0.
Now since w is determined by any algebraic equation as a
continuous function of z, round each point can be assigned a finite
region throughout which g(z, Wi), and likewise h (z, Wk), each regarded
as a function of z, has the value zero. But hence follows that each of
these functions must be zero in the entire connected n-leaved surface;
for, the function can be extended out from the finite domain into
each leaf by means of the expansion in series of positive integer
powers. But since by hypothesis wi can be carried over continuously
into every other value of the function, we have therefore:
g(z, w1) = 0, gS, w2) = 01... g(z, Wn) = 0,
which is only possible when g is of the nth degree with respect to
w. When the sum of the infinitudes of the function w is m, g must
also be of the mth degree in z, i. e. the factor h (z, wk) is of the order
0 with respect to z also. We have therefore:
f'(z"', w1) = Const. g (z', w,). Q. E. D.