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.

~ 188. Successive derivates of the implicit algebraic function.

349

Successive differentiation of this quotient, a calculation without
difficulty but leading to long formulas, presents as many derived
functions as we please. It is essential that in all of them the
only quantity occurring in the denominator is -, which does not vanish
as long as we keep in a domain of regular points, so that the values
of all the derived functions are finite and determinate, since the points
are excluded in which w becomes infinite.
Although we have thus indicated that which is essential in the
equations whereby the derived functions are calculated, it is still
important for a subsequent application of the formulas, to introduce
some abridgments of notation which give us a comprehensive view
of the equations that are to be formed."^) Let us denote the
41w
derivate -d by Wk, the partial derived function by fk —p,
and let us write:
p-k
kp=
w - w0
then, for the ratio - a = w, which for S a becomes the derivate
2 - Ca
dw
d4t we have the equation:
o -= 1t + ( - c)0t2 + (v - c)2' a..
ascending by powers of z to the term (z - a)'l or (- - a,) according
as m is greater or is less than n.
From the equation w - Wa,'= wl(S - a) we find that:
Cdk2 d +iv, d4k-1 l4 = k -a ) dZ + z1;
so that for - = a we have the relation
dzk dzk-1
a
Accordingly the higher derived functions w2, w3, etc., are found by
differentiating the above equation totally with respect to z, and putting
= a, d k1 ' * t Wk in the derived equations. Only the terms
up to the power (z - a)k- in the equation are required in establishing
the recurring formula for the /cth derived function. For the lowest
values of k we find the following equations:
*) Plfcker (1801-68): Theorie der algebraischen Curven, p. 156. Bonn 1839.
Liouville (1809-82): Memoire sur quelques propositions generales de geometric
et sur la th6orie de 1'dlimination dans les 6quations algdbriques. J. de math., T. VI.