procedure is to find first the non-normalized autocorrelation 
function and then to determine [ac )]° from it. 
The non-normalized autocorrelation function is given by 
Q(p), where p is a continuous variable, in equation (10.26). In 
the first expression on the right in the top line of equation 
(10.26) Q(p) is defined in terms of 7(t), the free surface as a 
function of time, as observed at a fixed point. The variable, p, 
has the dimensions of time, and 7(t + p) is simply the value of 
7) which is found p seconds after the time, t. In the second ex- 
pression, the Lebesgue Stieltjes representation for 1) (t) is sub- 
stituted for 7(t). On the next line, the integral is represented 
by the limit of a partial sum. The results are equally valid for 
the non-Gaussian cases discussed in Chapter 7. 
For q not equal to n, the product term which results can be 
written as the sum of two trigonometric terms which when averaged 
over time average to zero, and consequently the expression simpli- 
fies to the third expression. Upon rearrangement, the fourth 
expression is obtained in which the square of the cosine term 
has a net positive mean, and the cosine sine term averages to 
zeroe The fifth expression extracts the constant part of the 
cosine squared term; all other terms are sines and cosines, and 
in the limit, average to zero. Integration over time yields the 
bottom expression on the left, and, in the limit, the last two 
integral forms are the result. These integrals are ordinary 
Stieltjes integrals without the square root sign as in the power 
aehcereaveulse ihe [acu )]° is a piecewise continuous function, the 
last integral is an ordinary Riemann integral (i.e. the kind one 
can often look up in tables of integrals to evaluate). 
=) 7 
