the last sinusoidal term of the integrand, the integration of 
equation (8.48) would then yield the final result. Note that 
as De and We approach infinity, the application of the lemma 
given in equation (8.9) yields a simple sine wave of the form 
A sin((u 4°/g) (x cose, + y sin®,) - Ht) without edges. The 
integration as it stands, for D, and W, finite, is too difficult 
to carry out and it must be approximated. The term involving x 
in the last sinusoidal term of the integrand is approximated in 
equation (8.49). The second expression in equation (8.49) is 
simply a way to rewrite the original expression. Since the major 
contribution of the integral occurs near a and 8B equal to zero 
from the behavior of the other terms in the integrand, higher 
order terms such as those involving a> and at can be neglected. 
The third expression in equation (8.49) employs this approxi- 
mation. Also since the major contribution is given near a and 
B equal to zero the square root can be approximated by the first 
term in its binomial expansion, and the fourth expression is ob- 
tained. The final expression in equation (8.49) is the result of 
clearing fractions. 
Equation (8.50) is the aippioeinate result which is obtained 
when the approximation given in equation (8.49) is substituted 
into equation (8.48). The first term in the argument of the last 
sinusoidal term of the integrand is simply a constant as far as 
the parameters of integration are concerned. The remaining terms 
are functions of ae. a, Bo and B alone without cross product terms 
of the form of, say, a8. 
For simplicity let /1 - pie = K as defined by equation (8.51). 
= 194 = 
