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.

~ 200. 201. Simplest case of the many-elementary point. 379
- (fk,O + 7,i-ktw, + * * ipfk-p,PU0 + *. + fO,ktUl) = k = 0.
By our hypothesis the k roots of this equation are determinate
quantities not infinite; we shall further assume, that they are also
all different.
In the k-elementary point there is then no branching;
for, all the higher derived functions, belonging respectively to the
various values of w, remain finite. These are to be deduced from
a system of equations which are obtained, in analogy with those
established in ~ 188, by successive total differentiation.
Denoting the quotient w- by w1, because Lim (-) w1 = i'
we should have in general:
dnw dnow din-, d-I 1 dn
-= Z - + n - therefore when z = 0:
d n d z' dn-1 ' dzn- 1 d dz'
Now the equation for w1 arranged by powers of z is:
%k + z -+l + Z P+2 +- ** 0,
and from this we obtain for determining the successive derivates at the
point z = 0 the equations:
2 k + k+1 = 0,
W3 aak, ( 2 a2', a O
I aWt, 2+ 2) t-12 + t2 + k+2 0;
W4 a Ok +3 3 aa k+l W2 20 k
U!i wl biT 2 awl2. (a k+2 +2 1 aQkil (W022 i aa Ok 2\3,,^ ~ ~...., 1. W,,.....3, 2,.
O\ w1 2+ 2 zw2 (2 'i ) + W1 (,y )} + +3 0.
These equations present successively finite determinate values for
w2, w3..., since the factor -k does not vanish. Corresponding to
each different value of w, they give uniquely a different value of each
higher derived function, and accordingly lo different expansions.
The result is formulated in the theorem:
When in the c-elementary point all the values of the first derived
function are finite and different, k elements of the function meet in
this point and each of these is only a simple element, i. e. each can be
expressed by a series of positive integer powers.
When the point considered is real, and the k values of the first
derivate likewise real, lc branches of the algebraic curve with distinct
directions of tangents pass through the k-elementary point, and each
right line through it w = ao has in it at least k points common with
the curve; it is then called a multiple point of the order k7 without
ramification.