The first step is to find the Fourier Spectrum of cos a°x/¢ 
as a function of a. Equation (5.9) gives this spectrum as a func- 
tion of the new variable, 8. It then follows that cos a°x/g is 
given by equation (5.10). Equation (5.11) would be the correspond- 
ing equation for the sin xa°/g. Note that this step involves the 
assumption that x is greater than zero. Slight modifications of 
the analysis from this point on would also yield valid results for 
x less than zero. 
In the second expression in equation (5.12), (Plate XI), equa- 
tion (5.10) has been substituted for cos a-x/g in the integral 
which occurs in the first term of equation (5.8). In the third 
expression, the order of integration has been interchanged as in- 
dicated by the rearrangement of the brackets. The change in the 
order of integration can be justified théoretically. The term 
in brackets in the third expression leads to the conclusion that 
integration of the term, cos 8° e/4x + sin 8° e/Ax, from zero to the 
indicated variable upper limit is equal to the original integral as 
in the fourth expression. Finally a change in variable gives the 
last expression in equation (5.12). 
The term in brackets in the third expression in equation (5.12) 
is considered in equation (5.12) and designated with the letter re 
It can be shown easily that this integral as a function of 8, after 
integration over a, is either constant or zero, and its values are 
given below the integral (see Pierce, [1929]). 
Now that the integral over a has been evaluated, consider the 
integration over Bp when 47x/gT - t + nT>0. For 8 <4rx/gT - t + nT, 
the integrand as a function of 8 is equal to cos B°2/4x ne Salil Be /4x 
= B82 Oo 
