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.

~ 57 —59.

The most general algebraic function.

103

which z = 0, there is also a differential quotient identical when taken
progressively and regressively, to be calculated from the equation:
Of
d y a x mnB - - xm -1 +t (m - 1) B x- +. + B,-i
dX af Ay-'- + (n - i) -Al n-2 + 1+ An —1
ay
and that becomes infinite at the points for which the denominator
Of
vanishes, where simultaneously f- 0 and =-= 0.
The preliminary result is therefore: If in the algebraical equation
f(x, y)= 0, y be considered as a function of x, this function has at
each point a differential quotient; in other words: it can be carried
on continuously from each point. Geometrically stated this is the
theorem: an algebraical curve has at each point a tangent; it cannot
break off at any point.
This theorem however undergoes modifications: for there can be
points, at which numerator and denominator of the quotient d vanish
simultaneously; or simultaneously increase beyond all limits; these
require a special investigation.
59. The Theorem of the Mean Value in its most general form
(Taylor's series) shows directly whether a function can be carried on
continuously for finite values of x and y. We found ~ 56 that the
value of f(x + h, y -+ -) can be calculated from the equation
Of)-+ 202f i af f +]2 02f/'
tf(x+h, y+)=tf(x,y) + I< f+k af + h X+ 2h U 2 ~
1n-o
1 ay a + OX 7 Y+ a
' ' ~' -/ ~- o lx ]Y' fy(x+Oh, y - Ok)
— p= oax'-v aYP
We commence fiom a point x0, Yo, at which f= 0, and try to
find another in its neighbourhood, i. e. for arbitrarily small values of
h and k, at which f(xo + h, Yo + k) likewise vanishes. Denoting
the values of the partial derived functions at x0, y0 by ()0, the values
h and k must satisfy
h0 -{1"(af) + lk ++ + { -() +axk+/o aay/ yI+2() +, ) 1 + (f
"+ 2 i hm p xn ( a yp )
Since arbitrarily small values of h and k are in question, we see that:
af a f
unless and vanish simultaneously, terms containing higher
powers of h and k are arbitrarily small compared with terms of
the first dimension; thus the continuation of the implicit function is