infinite. However, P.E. as defined by equation C7226) nies 
still finite. The appropriate graphs for example four are given 
in figure 13. In this case, there is no jump in E(u), and 
dE(y)/dp equals a if p is less than 27/T). 
The Gaussian case, or the principle of independent phases 
The examples of the integration of equation (7.1) which have 
been given so far have not introduced anything basically new in 
the nature of the sea surface. The integral is so general, how- 
ever, that it includes many cases which can only be represented 
by such an integral. One special case is the Gaussian case 
which is of extremely great importance in the theory of noise 
and which will prove to be of equal importance in wave theory. 
The integral considered is still equation (7.1), and the 
conditions given by equations (7.2), (7.3), (7.4), and (7.5) 
are still imposed. In addition, the condition that E(p ) be 
a continuous function is added, and the point set which defines 
W(#) is very specially defined. Continuity in E(p#) yields all 
necessary qualities. It permits a very peculiar mathematical form 
for the derivative of E(#) which will be discussed later in 
Chapter 10. 
Continuity of E(u) is imposed by equation (7.27), which 
states that the difference between E(y 5,5) and E(y 5,) can 
be made smaller than some delta if 5,5 - pop is made smaller 
than some epsilon (which may depend on delta). * In examples 
one, two, and three, this condition is not fulfilled at the 
jumps. In example four, E(w) is continuous. 
*See Chasant [1937]. 
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