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.

174

The definite and the indefinite integral.

Bk. III. ch. I.

into subdivisions; let S' denote the corresponding sum of products
formed like S by multiplying each new partial interval by the value
of f at the beginning of that interval; let n' be the number of the
new intervals; after this each of these intervals is to be broken up
into an arbitrary number of subdivisions, let the respective value of
the sum of products be called S" and the number of intervals n";
proceeding thus we obtain a series of arbitrarily increasing numbers:
I, in,... (k)... etc.
and a corresponding series of sums:
S, S', "...k S() etc..
This series must represent a determinate limiting value, i. e. for
any number 6 however small, it must be possible to find a value n(k)
such that the difference between S(k) and any following value S(k+v)
shall be smaller than 6.
We first remark, that a sum of the form S can be represented
always by an expression of the form:
III. S = (x - a) f (a + O(x - a)), where 0 < 0 < 1;
for if, taking account of the sign, the greatest value among the
coefficients f(a)... f(Xn-,) be denoted by G, and the least by K,
we have K(x - a) < S < G(x - a), or S is equal to the product
of x -a by a value between K and G. Now because f(x) is a
continuous function of x, it assumes at least once each value between
the least value K and the greatest G, it overleaps none, therefore there
must be a point at which f actually has the value that is requisite for
equation III.
Now if each of the intervals from a to x1, from x1 to x2, etc.
be divided into smaller intervals, new sums come up in place of the
products d1f(a), d2f(x)...; namely, when the dividing points in
the kth interval: from xk-_ to Xk, are denoted by x,(k) x2(k)... x-k)
the product dkf(Xk-l) is replaced by the sum:
(k).== - (x,(^)) -XI _ 1)f (xkl)+(x2) -— xI1 () ) f(x (k))+ Xk - XV..- I (k) )f (Xl(k))
In analogy with equation III. the sum on the right can be brought to
the form:
(k)
= (Xk- xS1) f (Xk -1 + 0k (X1 - X-)) - kf (Xk-1 + Ok (k- Ok-1))
O < Ok< 1.
Thus, partition of the intervals of S into new
subdivisions leads to a value S', that only differs from
the former by each term dkf(X-_l) containing in place of