of R vectors of unit length all pointing in randomly picked 
directions. Let the sum be the complex vector given by B pexp(io,). 
The other sum is a sum of R vectors and every vector in this sum 
points in exactly the opposite direction to the corresponding term 
in the first sum. The sum is therefore of the form B pexp(-ie,) 
and it points in exactly the opposite direction. The complex pro- 
duct is therefore always a real positive number (Bap) as given 
by the last expression in equation (10.14). 
From equation (10.13), the sine of W (j) can be written in 
the various forms given in equation (10.15). The results of equa- 
tion (10.14) permit the use of terms like Bypexp(iese), and in 
this case the Bar cancel out. The sum therefore represents the 
Sine of some angle, 5R° But from the nature of the sums dis- 
cussed in the paragraph above, the value of OR for a large value 
of R is equally probably any value from zero to 27. Equation 
(10.16) is therefore the result, and the probability distribution 
of W'(j) is the same as that required originally in equation (7.28). 
These results are next substituted into equation (10.17). 
Since the values of the Bir are not one, one begins to suspect 
that things are getting complicated. Also the results are be- 
ginning to look something like the results which were obtained 
in Chapter 7. 
So far in the proof only one small strip between pz 2k+25 
and pu Det25+2 has been treated. There are N of these strips 
between # » and # y,5- The values of AE, can, by picking the 
points p 2k+24? all be made equal to 
[E5( byyoet/2) - E5(H 4.7/2) ]/N as shown in equation (10.18). 
= 26a" 
