The terms, xK, * Yer and =p x+ Ky, occur as units in the inte- 
grand. The term, xK, + YPyz> is the direction in which the wave 
crests travel, and the term, -p,;x + Ky is the direction of 
orientation of the wave crests. They are the coordinates of 
a rectangular coordinate system which has been rotated through 
the angle 8,, and they are designated by X and ¥ in equations 
ik 
(8.52) and (8.53). Note that X and ¥ are perpendicular. 
The constant term with respect to the variables of inte- 
gration can be factored out by a trigonometric identity, and the 
use of the notations given above them yields equation (8.54). The 
notation for the various constant terms can be shortened by the 
use of the symbols, C, D, E, and F, as defined by equations (8.55), 
(8.56), (8.57) and (8.58). Then the trigonometric terms under 
the double integral can be split into a product of two integrals 
by expanding them by a trigonometric identity and the result is 
equation (8.59). 
Each of the integrals in equation (8.59) is an integral 
over only one variable and if one of them can be evaluated, all 
can be evaluated by similar techniques. The integral of one of 
integrals is given by equation (8.60). It is integrated by the 
very same techniques that were employed in the integration of 
a function of similar form in Chapter 5. The steps from equations 
(5.9) to equation (5.12) in Chapter 5 could be carried out (with a 
different variable for the notation) in order to obtain equation 
(8.60). 
The integration of equation (8.59) would then result from 
the substitution of equations like (8.60) into equation (8.59). 
Delta is the dummy variable of integration for thos® expressions 
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