1910-11.] On Fourier’s Repeated Integral. 573 
Cor. — When the upper limit of integration with respect to v is finite , 
we may reverse the order of integration. 
For, as in the proof of the preceding corollary, 
j>J, g{u) cos u(y 4- k)du = +k dv\ q g(u) sin uv du - J* dv\ q g(u) cos uv d,u, 
and, by the theorem quoted in the proof of the present theorem, we may 
change the order of integration in each of the integrals on the right, and 
therefore we may do so in the integral on the left. Similarly we may do 
so in \*dv\ q g(u) cos u(v — k)du, and therefore in Jo \*g( u ) sin sin u v du. 
§ 15. Lemma . — If the series of proper or improper Lebesgue or Harnack- 
Lebesgue integrals, 
\ Z q fl( X ) dx + \lM X ) dx + • • • • +\ Z q fn( x ) dx + • • • • 
converge to the proper or improper Lebesgue or Harnack-Lebesgue integral 
\\.f{x)dx, 
for all values of q and z in the completely open interval (p<q<z<B), in 
such a manner that a convergent series of positive quantities, 
+ c 2 4- . . . . + c n 4- . • . • , 
can be found, each term of which is not less than the absolute value of the 
corresponding term of the series of integrals, whatever be the values of q 
and z in the interval in question, then, provided only in addition 
\lfn( X ) dx 
exists, for each value of n, we can assert that 
(i.) Jpf(x)dx exists', 
(ii.) the series 
\pfi(x) dx + \lf 2 (x) dx + .... 4- \lf n (x)dx 4- .... 
converges ; 
(iii.) the sum of the latter series is equal to the former integral. Here the 
limits of p and B may be either finite or infinite. 
We first prove the lemma when the lower limit of integration is q 
instead of p. 
For since, by hypothesis, | \ z q f n (x)dx | L c n , and \*f n (x)dx exists, so that it 
is the unique limit of the former integral when 0 approaches B 
| \*f n (x) dx | Lc n . 
Hence, by Weierstrass’s test for convergency, the result (ii.) follows when 
for p we write q. 
Again, it immediately follows from the premisses that 
\\ z q f(x)dx - \ z q f{x)dx - \ z q ffx)dx - .... -■\ z Jn{ x ) dx \Lc n+l + c w + .... 
