254 MATHEMATICS: N. WIENER Proc. N. A. S. 
ber of values of the argument of /—if F {/} is a function 3 of f(h), /(&),.,., 
f(t n ) of the form 3>[ /(/„)] — then it is easy enough to give a 
natural definition of the average of F, which we shall write A {F}. 
We can reasonably say 
_Xl* _ (X2-Xl)i (^ W -X W _ 1 )» 
e h '«-<»- 1 dxi...dx, 
In particular if F{/} =1/(01^ 1/(0.^ - [/(« P*. 
then 
+ 
+ "« * 'n-'n- l dy 1 ...dy H 
This latter integral is in the form 
/00 /*°^ 2 
... f P(0i,... t 0 w >~^ z «dz h ...dz n 
—00 ^ —00 
where P is a polynomial, and can be evaluated by means of the well known 
formulae: 
f e- yi y 2n + 1 dy=0, 
V — on 
f e- y *y 2n dy = yJ 
_ 1.3.5...(2n-l) 
7T 2 n 
We can thus easily evaluate A {F} as a polynomial in h, fa s J M , which we 
shall call P Wj , mn (t u ...,t n ). It is easy to show that if Sm ft is odd, 
To return to the more general functional : there is a large class of so-called 
analytic functionals, 4 which may be expanded in the form of series such as 
