zero. That is, the integral is the limit of the process defined 
by equation (7.7) and the law for picking YC bony) as the 
greatest distance between two successive / 's in the net, say 
H+] and ,, is shrunk to zero. Note that the partial sums 
in equation (7.7) are almost periodic functions as defined by 
Bohr [1947] if the 4's are irrational. 
This integral has one very valuable property. The square 
of the function given by the integral averaged over time, is 
equal to (1/2)E. ax: This can most easily be shown by consider- 
ing the partial sum given by equation (7.9) in which the small- 
est distance between two successive #'s is Aj 9 a Small but 
finite value and in which the largest distance between two suc- 
cessive # 's is A,p#.* The potential energy of 7 (t) averaged 
over time is given by the integral expression in equation (7.10) 
(see equation (3.10)). Since the 4 5,,, are different, the 
cross product terms in the Square average to zero when (7.9) 
is substituted for 7(t), and the second expression in (7.10) 
results (see, for example, equation (6.22)). Upon rearrange- 
ment and evaluation of the sum, the plus E( #5) in the first 
parenthesis is cancelled by the minus E(w.) in the second 
parenthesis, and the r'th partial sum is EC Hongo) As f ap=- 
proaches infinity, P.E. equals (pg/4)E,.,. Equation (7.10) 
holds for arbitrarily small values of Ask and hence it holds 
in the limit. 
*A, b is the smallest segment in the net; Ap is the largest. 
The lengths of all others lie in between. 
= 129 = 
