258 MATHEMATICS: N. WIENER Proc. N. A. S. 
where the total variation of each \p n is less than some quantity independent 
of x, and let each \p n be uniformly continuous in x over the interval (a, b) . 
Let u(%) be a function of limited total variation in x over the same interval. 
Let A { flF x du } exist, and let 
Urn 2 f 1 ... f A { f(xi) ...f(x n ) } dt n fe, x n , x)=0 
w = oo n = m ^ 0 •/ 0 
uniformly in x. Then 
J* A{F x \du = a| j b F x du}. 
As to (5), let us begin as above with a functional of the form 
$ { /} = j[ - f Q f(*i) .../(*») *„). 
Consider 
I = f\ - J] A {/(*!) ...J(x„] dk. (x u .... x n ), 
where A is taken in the original sense as an w-fold integral. By definition 
I = Urn y. A {/(?,„) .../(?«,)} A^+J + V„(*i, ..,.*.), 
*ia. 
0 
where % = 0 ) %,... ) ^ (l =l is an increasing sequence of numbers, lies 
between ^ x and tf^K+i and Km is taken as max(x ktK+1 —x kK ) approaches 
0. Let V be the total variation of \f/ n as its arguments range from 0 
to 1, and let M stand for max(x k)K+1 —x kK ). Let Q stand for the least 
upper bound of the variation of A {f(xi}... f(x n ) } as the point (x h x n ) 
wanders over an interval ( * 1,a+ i • - Xn *+ 1 ) Then 
(6) i - a i 2 -m«*)C^ + x n ; x "' * + 1 • •> *») } i ^0- 
a,...,t? ^ 
Now, let be that function whose graph is the broken line with 
corners at {x (<K) ,f{x^ K) ), where OJ^KjSji, # (o) = 0, # (m) = 1. Then if x lies 
between x (K) and x (K+1) ,f m (x) is of the form aj m (x {K) ) +bf m (x iK) ) +a+b. 
It follows that if (£i a , .... lies in the interval f^"* 1 ) V 
^ {/mfeJ-./m^n,?)} is of the form 
flid + OtPi + ... + ApCp 
ai + 02 + ... + a p 
where each C fe is the value of some A {/(£«).../(£,,,>) } such that 
is a corner of the hyperparaUelopiped(^ a ) +1 ^^ (,?+1) ^. It readily results 
from considerations of continuity that A {/ m (£i).../ w (£ M ) } is of the form 
A {Kv ia )-..f(vn») } , where n n& ) also lies in the interval Y 
