given by equations (9.41) through (9.45) are added, then the 
limit of the partial sum defined by equation (9.22) still exists, 
and the integral given by equation (9.1) still has the property 
given by equation (9.24). 
If E,(,) has a continuous second mixed partial derivative, 
it must be everywhere greater than or equal to zero. Consequently, 
it can always be written as the square of some function An(H 58), 
and under some conditions equation (9.46) is another way to write 
denn GH 40). Substitution of equation (9.46) in equation (9.1) 
yields equation (9.47) which has a meaning only in the Gaussian 
case. The expressions for the partial sums can then be written 
in the forms given in equation (9.48). In the last expression 
in equation (9.48), the partial sum has again been expressed as 
a vector in the complex plane. It will be shown in Chapter 10 
that for a fixed x and y as t varies, the short crested sea sur- 
face as observed at a point has all the properties studied in 
Chapter 7 for 7(t). The exact relations between E,(H ,6), E(u), 
(Aj(p 4e))? and [a(p )]° will also be discussed at that time. 
Some examples of cumulative power density functions and their 
power spectra 
Values of [Ay (m ,e)]? have never been obtained for an actual 
sea surface because the observations needed on which the compu- 
tation of the function depend have never been obtained. Some 
examples of what the function might look like can be given, and 
then the consequences of the form of [Ap (nu ,e)]° in the results 
of a hypothetical forecast can be described. It will be seen that 
the nature of the forecasted values depends critically on the 
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