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.

312 General theorems concerning the double integral. Bk. III. ch. VIII.
even infinite; we call the function then "in general" continuous. In
the last case if the function become determinately infinite it must be
algebraically infinite at each isolated point in a lower order than the
second and along entire curves in a lower order than the first.
For each value of y then ff(x, y)dx represents a function of x
also in general continuous and finite, that is integrable with respect
to y. Denoting:
f(x, y)dx -- F(x, y) - F(a, y),
we have:
f (x, S) a F(x, y)
Each parallel to the ais of x must cut the boundary curves in a
Each parallel to the axis of x must cut the boundary curves in a

jr

a - a.3, a
_
aQ 2
2 a
111
11~Ir a,

finite even number of points.
In our figure it is two or
four. Denoting the values of
x at the entrances and exits
belonging to a definite value
of y by x, x2, x3, 4, the double
integral:
f (x; ) dx d, is equal to
dy J'(, f( y)d x,
if the integral respecting x
x in this successive integration
be extended, for each value

Fig. 16.

of y, between the limits from xl to x2, and from x3 to x,. But in
the above notation:

Jf(x y)d x + f (x, y) dx F(x2, y)-F(x1, y)+ -IF(x4,y) -- F(x3 y)
Xi Xj
Therefore:
Jf(xy)dxy dyFxy) - lyF(xy)+- dyFx(IXy) fdIyF(x., y).
In these integrals the values of x as functions of y corresponding to
the equations of the boundary curves are to be substituted and then the
integrations with respect to y to be effected between the extreme values,