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.

Fourth Chapter.
The implicit algebraic function.*)
91. The most general form in which a variable w is defined as
an algebraic function of the variable z is by the vanishing of a
polynomial consisting of integer powers of z and of w:
1) f/(W, m) = Wo() Po (z) + ^1 a] (-) + -- W p n-rL( +) n n() = 0,
the factors p (z) being integer polynomials of arbitrary degree in 2 with
complex coefficients; let the highest power of z in any of them be m.
This form is of the nth order in w, assuming that all coefficients in
the polynomial q0o(z) do not vanish; let (q0 be of the degree 7k in a,
= 0 denoting that rq is a constant. It may be assumed that the
form 1) is not reducible, i. e. that f cannot be resolved into products of
algebraic expressions of lower order; for if it could, each factor equated
to zero might be investigated separately.
Integer and fractional rational functions are included in this
form, for these n - 1; in like manner it includes the explicit irrational
function treated above:
p
w = (z- a), which in form 1) is: w - ( - a)P = 0.
To each value of z, correspond n determinate values of w, different
or equal, the roots of equation 1), as was proved in last Chapter.
Let these be denoted by w1, w2,... Wn; they will vary according
to the value of z. The equation 1) presents therefore n functions
of 2, or in other words: it determines an n-valued function of z. The
following investigations have to demonstrate how these n branches
of the function may be separated, and how far they are continuous
functions with determinate derivates.**)
92. If the function w be considered at a determinate point, and
so one of the possible values w calculated for a determinate 2 == 0,
*) In this chapter the theorems of Algebra regarding the resultant and
the discriminant are supposed known.
"*) Cauchy: Exercices d'analyse et de physique math6matique. Tome II.
V. Puiseux (1820-83): Recherches sur les fonctions alg6briques. Journal de
Mathdmatiques, T. XV et XVI. 1850-1. (German translation by Fischer, Halle
1861). Briot et Bouquet: Th6orie des fonctions elliptiques. Paris 1875.