52 
IMPROPER MULTIPLE INTEGRALS 
If 1°, 
iim X 
/3=00 •'8 |B 
or if 2°, 
Mm T I/I 
6=00 
exist, then, 
7» 7* 
lim J /=J /- 
a, 6=» »¿/la, ft Vi21 
(1 
For, if 2° holds, 1° holds also, since 
asSt ^ 4 »- 
p 
Thus case 2° is reduced to 1°. Let then the 1° limit exist. 
We have 
J 9 ~J h ' 
*A)r „ »/21 „ «A>r „ 
*a/3 
a/3 —* 
as 4) in 44, l shows. Let now 
T) a (3 == A ab — 2l a /3- 
Tl ' el1 ’ / f </ *</ f 9- 
-JaJ JqJ 
(3 
But !D aj3 = 0, as a, /3 = oo, by 54. Let us now pass to the limit 
a, /3 — oo in 3). Since the limit of the last term is 0 by 53, 54, we 
get 
lini I g = lim I 
a, /3=co *^2l a £ a, 6=cc .4a, ft' 
9' 
Similarly, 
lim I h = lim I h. 
a, /3=oo V_*L/3 a, 6=oo iT^aft 
(4 
(5 
Passing to the limit in 2), we get, using 4), 5), 
f /= lim g- f A | 
J% a / a, 6=co (‘Liaft iLiaft I 
= lim f f 
a, 5=oo A ab 
In a similar manner we may establish 1) for the lower integrals.