ITERATED INTEGRALS 
63 
Iterated Integrals 
70. 1. We consider now the relations which exist between the 
integrals — 
(1 
and 
where 51 = 53 • (5 lies in a space 9? m , m = p + q, and 53 is a projection 
of 5i in the space 9f p . 
It is sometimes convenient to denote the last q coordinates of a 
point x — (x x ••• x p x P+1 ••• x p+q ) by y 1 ••• y q . Thus the coordinates 
x x ••• x p refer to 53 and y 1 ••• y q to (£. The section of 5i correspond 
ing to the point a in 53 may be denoted by (H x when it is desirable 
to indicate which of the sections (5 is meant. 
2. Let us set — 
(3 
then the integral 2) is 
It is important to note at once that although the integrand / is 
defined for each point in 51, the integrand <£ in 4) may not be. 
Example. Let 51 consist of the points (x, y) in the unit square : 
m n . 1 
x = — , 0 < y < —•• 
n n 
Then 51 is discrete. At the point (x, y)in 51, let 
Then 
X /=0 by32 - 
On the other hand 
for each point of 53. Thus the integrals 2) are not defined.