Voi,. 7, 1921 MATHEMATICS: N. WIENER 257 
and yp n is a function of limited variation in x h ...,$„. What we wish to 
prove is that 
f{f r f L A {/(*) -/fen) }#nfel, *») } <*« 
= A{ JV. JVfel) .../fen) fel,.-,^)} A* } 
Now 
a,..., t? 
If in this latter expression the total variation of \p n is less than a quantity 
independent of %, we can permute the f h a and the Urn, and get 
•6 
or, 
which we may write 
f Q '~f/( X d ~'f( X n)d[j ^«fel, ...»***"*) dttj. 
In this we suppose / uniformly continuous. It is easy to show that on 
our assumptions J*fy n (xi, ...,x n ,x)du is of limited variation. Conse- 
quently we obtain 
A { S { JV'JV^ •r/fe«)^«fe 1 ' ->**>*) 
=^{/]-//fei) •••/(*«)<* [//"fe 1 ' *«» } 
= f Q 'f Q A {/fed -/fen) }^ [j^ ^nfeb %n> *) <*ttj. 
A further transformation just like the preceding turns this into 
S {So'" So A ^ Xl ^ -/fe^ 1^" fe 1 ' **' ^ } ^ 
so that we have now a sufficient condition for the validity of our theorem. 
The extension to non-homogeneous terminating analytic functionals is 
obvious. The extension to non- terminating analytic functionals may 
be deduced with the help of (3) and a well-known theorem on the in- 
tegration of uniformly convergent series, and reads as follows: let F x , 
be an analytic functional of the form 
a 0 + s f f /fei) .../fe») #nfei» *«> *) 
l •/ o •/ o 
