NTEGRALS 
PROPER INTEGRALS 
8 
'or many purposes this 
form an unmixed divi- 
•e shall call unmixed di- 
¡ circumstances we have 
3, viz : 
1. Let A denote an un 
ióte those cells of £3 con- 
1. 
51. Let f(x x • • • xf) be 
on of 53 of norm 8 into 
the maximum and mini- 
\fd% (1 
at 
* f<M. (2 
a 
smonstrated in a similar 
manner entirely analo- 
S<8 0 . (8 
¡place S t , 8', 8 lK , by their 
ct that A is unmixed, to 
S<¿> 0 , ( 4 
t E be a cubical division 
so small that 
(5 
The cells of E containing points of 21 fall into two classes. 
1° the cells e lK containing points of the cell 8 L but of no other cell 
of A; 2° the cells e[ containing points of two or more cells of A. 
Thus we have _ 
S E = ^M iK e lK + ^M[e[, 
where M lK , M[, are the maxima of f in e lK , e[. Then as above we 
have 
S E < e <e 0 , (6 
if e 0 is taken sufficiently small. 
On the other hand, we have 
\S A - 23JU, 
KF^-ZeJ. 
Now we may suppose S 0 , e 0 are taken so small that 
SSo 2e LK 
differ from 21 by as little as we choose. We have therefore for 
properly chosen S 0 , e 0 , 
This- with 6) gives 
which with 5) proves 4). 
4. Let f(x x • • • xf) be limited in the limited field 21. Let A be 
an unmixed division of%of norm 8, into cells 8 2 ■ • •. Let 
£ A = 2mA, S a ='2M 1 8„ 
where as usual m L , M i are the minimum and maximum of f in 8 t . 
Then 
C fd% = Max Sto f fd% = Mi 
Min 
The proof is entirely similar to I, 723, replacing the theorem 
there used by 2, 3. 
5. In connection with 4 and the theorem I, 696, 723 it may be 
well to caution the reader against an error which students are apt 
to make. The theorems I, 696, 1, 2 are not necessarily true if f