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.

~ 166. 167.

Definition.

295

The enunciation of the answer is as it was for the simple integral:
If we denote the greatest fluctuation of the finction, i. e. the positive
difference of its greatest and least values in the element,, or at its
limits, by D,,; the sum: T1 D + r2D2 + + r'' Dn must converge
to sero along with the quantities r.
In the first place it can be seen that when for any one continued
process of partition this sum converges to zero, it converges likewise
to zero for every other. The proof is the same as for the simple
integral, only that the conception of the superficial element everywhere
replaces that of the linear interval.*)
In the second place we prove that the condition enunciated is
necessary, by starting from a determinate partition and continuing it
by resolving each element into further elements. The sums
formed with the greatest and least values of the function: G,, and g,
in the interval rv,
G,Er,, and 2g a
approximate in this process, the first by continued decrease, the second
by continued increase, each to a determinate limiting value, and these
two limiting values become equal only when we have:
2(G - g)^'r =2Dr, = 0.
In the third place it can be seen that the same limiting value is
obtained in another partition of the same kind, when two different
partitions, each already pushed so far as to yield a value differing
arbitrarily little from its limiting value, are considered simultaneously,
and this partition resulting from their combination is regarded as a
continuation as well of the one as of the other.
Finally we perceive that, provided the above condition is fulfilled,
we may also complete the process of partition without retaining the
limits of a former partition, because the series of values formed in
this way takes also a determinate limiting value, and each term of
this series ultimately differs arbitrarily little from the limiting value
reached by the previous process.
167. The necessary condition is fulfilled:
First: when f(x, y) is everywhere a continuous function (~ 52).
*) Here as in the case of linear intervals it is quite indifferent according
to what law the succession of the summands is formed. In that case, as in this,
since we are dealing with finite sums and want to demonstrate that it"is a property of such a sum to have a limiting value, there is no necessity that we
should take the intervals only in the exact order in which they are arranged in
the interval a to b. It is otherwise in the transition to the limit for integrals
in which the function to be integrated becomes infinite.