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.

104

Implicit functions.

Bk. I. ch. X.

indicated in the direction of its differential quotient
Of
k7 Ox
h ~ '
But now if and af simultaneously vanish', the first member of our
equation disappears, and since for the arbitrarily small values of h
and k for which we are enquiring, the 3rd, 4th etc. powers are arbitrarily small compared with the second, the limit of the ratio of k to
7I is to be extracted from the quadratic equation:
(,,x2 a(hk0 + i( ) =o 0.
( a 2o + xG SC G y( + (
When this equation is written in the form
7( J (^ \ IA ' If (~ \ T2f t
{ h o2 o \ +oy, I } = ) }C ( (
it shows, that the ratio - has two real values, different or equal, or
no real value at all, according as
{c ( y 2o}- (2f) (a2o
is greater than, equal to or less than zero. In the last case the function f (x, y)= 0 cannot be continued from the point xo y, in any
direction by real values of x and y; while in the first, two different
directions are found. Its curve has an isolated point or a double
point with real branches, singularities that can also occur in algebraic
curves. If (f), (fa) 7 (lf) ' also vanish, we are led to a cubic
\ax/o' xaxay 0 Y2 /o
equation which presents for the ratio of k: h either three real values,
different or equal, or else only one real value. Such singularities
can rise higher, but the further discussions require theorems as to the
number and nature of the solutions of equations of the nth degree.
These remarks contain only the first germs of a problem which may
be stated generally thus: The implicit algebraic function f (x, y) -= 0
is given. For x = x0, y takes the value y0. It is required to express y as an explicit function of x by a convergent series of powers,
subject to the condition, that the relation f(x, y)= 0 remain always
satisfied, and that for x = x, y-Y=0. But we must postpone the
solution of this problem until later on, for it requires a considerable
extension of our previous conceptions. In the first place we must
be able to tell, how many values of y belong to a determinate x,.
This requires the investigation of complex solutions. We must then
solve in general the question as to whether a function, in whatever