72 
IMPROPER MULTIPLE INTEGRALS 
But for a fixed n 
Jg 
Then 
is limited in 33. 
Hence for Cr 0 sufficiently large, 
r f < T/ , at each point of 33, Cr 0 < &- 
*dr 
where T № , y ra are points of T in (5 n , c„. 
Hence — — — 
LJr-LS n + Lo£n 
(6 
(7 
(8 
Now 33o may not be complete; if not let B G be completed 33«. 
As 33 is complete, 
LSj=.LJj- 
-%0—yn '—B Q— Y» 
We may therefore write 8), using 5) 
-«+№//+/ fzff+S /• 
Z«^g *"B«Zg„ '¿B ( ;'Ly n 
By 75, the last term on the right = 0 as n = go. Thus passing 
to the limit, 
f f/Clini f f /, 
C ( d 
since e > 0 is small at pleasure. 
On the other hand, passing to the limit Gr = oo in 7), and then 
n — oo, we get 
lim f f < f f 
(10 
Thus 3), 10), 9), and 4) give 1). 
Let us now suppose that the middle term of 1) is divergent. 
We have as before 
J J*< limj^ J* < J /. 
Hence the integral on the right of 1) is divergent.