of Continuous Functions. 
529 
word integration to define the integrals of the former as limits of 
integrals of the latter. 
It should be remarked that our definition shares with that 
of De La Yallee Poussin the disadvantage of applying only to 
absolutely existent integrals. The fact, however, that it employs 
only well-defined series of continuous functions, and that its form 
is independent of the number of dimensions, and of the character 
of the region of integration*", gives it certain advantages over 
that of Harnack. These advantages are best brought out in a 
systematic development of the theory of Riemann integration 
based on the new definition. Such an exposition of the theory 
I hope to be able to publish shortly. 
* It should be remarked that, when the region of integration is infinite, an 
upper and lower integral, as defined in this paper, will not exist unless the con- 
tinuous function <f> n>r possesses an integral over that region. This case requires 
further discussion. We might so modify our definition that the region of integra- 
tion is the whole plane, or S n , by ascribing, in fact, the value zero to the original 
function at every point at which it is not already defined. But this would not 
completely remove the difficulty, a continuous function has not necessarily an 
integral, finite or infinite, over, for example, the whole infinite plane, according to 
the usual definition. Even when the region is finite, if it be bounded by a curve of 
positive area, the definition requires care. 
Moreover even when all this can be adjusted, the extension of the definition of 
§ 7 to an infinite region of integration may still be unallowable owing to the viola- 
tion of the fundamental law that the integral over an infinite region S is to be the 
limit of the integral over a finite region S'. It is, in fact, not a priori allowable to 
change the order of the limits and write 
Lt Lt / </> n> ,, ( x ) dx = Lt Lt I <p n r ( x ) dx. 
r = oo S’=S J S' ’ S’=S r=o o J S' ’ 
Unless this is true we cannot assert that, defining the integrals as in § 7, 
