256 MA THEM A TICS: N. WIENER Proc. N. A. S. 
(1) A{F 2 } +A{F 2 \ = A{F 1 + F 2 \ 
(2) cA{F 1 \ = AlcF^ 
(3) ?A{F n } = A^F n j 
(4) If F x is a functional depending on the parameter x, and u(x) is 
a function of limited total variation, then 
f a A{F x }du = A{f a F x du} 
(5) Suppose Fh, ...,t n (x h x n ) be defined as F< f Xl '~" Xn (t) >, where 
h,...,tn ) 
f Xl '-> Xn (t) assumes the value 
h, In 
v _l (t - h) (x k + 1 - x k ) 
%k n 
h + i — h 
for t k <t<t k -v Then 
A{F} =limir-~^{t k -h- 1 )~ 1/2 j ...j F hi )tn (x u >..,%^ 
_xS_ * ( x k~ x k- i) 2 
e h 2 (tk — t k _ 1 ) dxi...dx n , 
where the limit is taken as the ^'s increase in number in such a manner 
to divide the interval from 0 to 1 more and more finely. 
2. The next task before us is to investigate hypotheses which are 
sufficient to guarantee the validity of propositions (1)— (5). Propositions 
(1) and (2) require indeed very little discussion, for they are always satis- 
fied when the series for Fi, F 2 , ^.{Fi} and A{F 2 } converge. In (3), let 
Fi,..., F„,..., and the series SF n all possess averages, and let 
00 m 
SF M =SF K + F m 
1 i 
where A {F OT } vanishes as m increases without limit. Then 
A^F n j =2A{F n } +A{R m }. 
Therefore 
Urn |a jsF w j -Sa|f„|| = 0 
and (3) is proved. If SF W converges and lim A{R m \ = 0, we shall say 
SF W converges smoothly. 
Proposition (4) reduces to the ordinary inversion of a multiple Stieltjes 
integral when F x {f} is of the form 
