equation (8.4) have both different directions and periods. Dif- 
ficulties arise when there are more than two terms in the sum 
and when the amplitudes are not the same. Under these conditions 
a sum of terms of different periods, directions, and amplitudes 
is the result such as in equation (8.5). The only method of 
analyzing the expression is to evaluate the expression term by 
term and sum them all in order to determine the actual appearance 
of the sea surface. There is no short cut to permit a form like 
equation (8.1). 
Equation (8.5) yields a multitude of representations for 
the sea surface, depending on the number of terms chosen to be in 
the sum. Figure 17 is an example of what the sea surface might 
look like with five terms in the sum of equation (8.5). The 
equation given on the figure was evaluated for t = 0 as a function 
of x and y. The contour system begins to look like some of the 
aerial photographs of the sea surface which are found in the 
literature. In equation (8.5). if -l<a,<1, and if the square 
root can have both positive and negative signs, then the expression 
is as general as possible. 
A_useful lemma 
In order to derive many of the results which ‘will follow, 
it is necessary to prove a very useful lemma (or auxiliary 
theorem* ) which has many applications in Fourier Integral Theory. 
This lemma will not be proved for the most peneral conditions 
on the functions possible. The conditions which will be assumed 
are general enough to include all of the cases in which it will 
*From lecture notes taken in a course in mathematics given by 
Professor Magnus of N.Y.U. Math Institute. Referred to also 
as Dirichlet's Limit formula. See Cy rant [1937] p. 321 for 
alternate proof. 
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