second, equation (8.20) is used. 
As M approaches infinity, the results of the lemma given in 
equation (8.9) can be applied. In the first integral, the range 
of integration includes a = 0, and in the second integral it does 
not. The limit in the first case is consequently a definite value. 
In the second case, it is zero. The second expression in equation 
(8.21) is the limiting value as M approaches infinity. 
The limit as N approaches infinity can now be studied. There 
are two terms in the bracket in the second expression of equation 
(8.21) and the integration can be written as the sum of two terms. 
The transformation of variables given by the upper sign (where 
applicable) in equation (8.22), (8.23), and (8.24) can be used in 
the first term, and the corresponding relations with the lower 
sign can be used in the second term. The result is the third 
expression in equation (8.21). 
The range of integration in both integrals includes the 
origin, and as N approaches infinity both integrals have a limit. . 
The limiting value is given by the last expression in equation 
(8.21). It is an even function in 9* as should be expected from 
the form of the original integral over 7 (o,y,t) times even cosine 
functions. 
In equation (8.25), a similar integral is evaluated where 
the cosines have been replaced by sines. The result is an odd 
funetion in O*. 
When the two equations (8.25) and (8.21), are taken together, 
it is possible to solve for a(# *,6*), and the result is given by 
equation (8.26). Similar operations with cos (m *)*/g sin@]sin p*t 
Sab = 
