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.

106

Implicit functions.

Bk. I. ch. X.

is not a finite value, when numerator and denominator simultaneously
vanish for x == o. This case shall be dealt with in ~ 62.
Now the same process holds good for a quotient of the form:
P (x(, y) in which, as in case of the implicit function, y is a function
9p (x, y) '
of x. This quotient must likewise be derived as limiting value from
neighbouring points, when for x = a, y = b, numerator and denominator vanish. We have
p (a, b) Lim p (a + h, b + 7)
p (a, b) h-=, k=o 9P (a + 7h b + k) '
and by the theorem of the mean value
rp (a + h, b +k) h 7 T,(a+ Oh, b + O c) a+ a (a + Oh, b + Ohk)
a a ab
C (a + h, b + k.) - h A (a + + 7, + ka, b + lk)
Accordingly
4sp. Lim
cp (a, b) a a ab ln 7Ib
'7^ (n, h) ^7a"Ap a;
p( - -,) 7-+ Linm
aa lb
If now in our case, in which p= af, q = a-j assuming
that the implicit function f(x, y) = 0 can be continued, we determine
the value of =d - by the same rule, remembering that by the above
proof Lim - = d, we obtain the equation
P h jdx
d2f f dy
dy z x2 axay dx o 2f (dy'\2 2 f dy if =0
dx 2'/' /' dy' or y, d) 'x 2 axay ' d- x2 =
axay ay2 ' dx
in which the values of the partial derived functions are taken at the
point under consideration. This equation coincides with that already
dy
found for h and k, and teaches that the quotient d-, provided it is
real, remains even in the singular points a continuous function of x,
for it can be derived as limiting value from neighbouring points. But
we may not take this second manner of calculating, as a proof that
such a thing exists, for here we are taking for granted not only the
existence of dy but also its continuity.
It is easily seen how this calculation adapts itself also to higher
singularities, the value of C(,') being developed on the hypothesis,
ip(a, b)
that all first derived functions, then all second, etc. vanish. We
do not here enter on the special case, that a and b are both infinite.