In the V 5995 plane for a fixed 0*, pick the net defined by 
equation (10.75) for vy, and (10.76) for 0,- Over the area element 
defined by equation (10.78) and (10.79), the value of [A, ( v 59030*) 1° 
can be designated by A,(h',j',j*) in equation (10.77). 
The net for the v.,0, plane for @* equal to 7/6 and for m 
equal to 10 and q equal to 3 is shown in figure 28 by the dashed 
lines. The circles shown by the solid lines show what happens to 
the lines # equal to a constant in the m,9 plane as they are 
mapped into the v,,@, plane. (See equations (10.48) and (10.49).) 
Consider what happens to the boundary curves which define the 
area covered by A,(h,j). The curve w equal to mw ,(2h + 1)/2m 
maps into v, equal to y, cos e(2h + 1)°/(2m)* as stated by 
equation (10.80). Similarly equation (10.81) is the mapping of 
# equal to #.(2h - 1)/2m. The straight line © equal to m(2j - 1)/4q 
maps into ©, equal to m(2(j - j*) - 1)/4q as stated by eauation 
(10.82). Similarly, equation (10.83) shows the mapping of 6 
equal to 7(2j + 1)/4q. Equations (10.80)and (10.81) are equations 
of circles which pass through the point v 6 equal to zero and 85 
equal to plus or minus 1/2. They are shown, for example, in 
figure 28. From these considerations, the area element A, (8,1) 
maps into the shaded area shown in figure 28. It therefore covers 
part of A,(6',0',1*) and An(7',0',1*). 
Power is conserved by the mapping. Since A, (hy §) is the con- 
stant value for the power spectrum over an individual area element, 
the integral over the area element in the # ,© plane is given by 
equation (10.84). After the mapping the integral over the area de- 
fined by equations (10.80), (10.81), (19.82), and (10.83) becomes an 
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