Vol.. 7, 1921 MATHEMATICS: N. WIENER 259 
Making use of this fact, and of the fact that 
l r°° r°° 
e h tn-tn-i dxi...dx n 
is an increasing functional of $(xi, ... : x n ), we can draw the conclusion that 
A{<i>{/ W }} = a{J\.. fj^xj ...f m (x n ) d* n (*i,...,* m )} 
if it exists, lies between the uppermost and lowermost values of 
^ A (X fy \ X fy \\ A *(*■+!) , / \ 
2j A {ftniixa) .../»(?»*) I \ a) xw *«(*!. *«) 
a, ...,t> 
and hence of 
From this and (6) we can deduce 
(7) \l-A {*{/„,} } | < 2VQ 
where Q is taken for Xuk = #Ck)- 
This proves our theorem for homogeneous analytic functionals. In 
precise terms, then, our general theorem will read: let 
00 00 
p if] = oo + 2 jj-f/w -Jfe) •■- x " ] - a ° + 2 F » w 
aw analytic functional. Let V n stand for the total variation of F n as its 
arguments range from o to i: Let (#( 0 ), ... , x^)) be a set of numbers in 
ascending order from o to r, inclusive. Let M stand for max (X( K+1) —X( K )) 
and Q n for the upper bound of the variation of A {f(xi)...f(x n ) } as the 
point (x h ...,x n ) wanders over the interval f^ a x +1) '"* ( * +l) Y Let /„(#) 
be the function whose graph is the broken line with corners at (o, 6) and 
(%{K),f(x(K))- Then if 
(a) Urn *y,V k Q k =0 
M = 0 
(6) A{F n {/ M } } exists for every and n according to the definition 
of A as a multiple integral : 
(c) the series for F converges smoothly; 
(d) 
multiple integral; it follows that 
A {F} = Urn A {F{f m }}, 
M = 0 
