(12.56). From the derivations of the power integrals involved, 
there is no correlation between 7 AI and 7 AID? and the two 
distributions are independent. These statements follow from the 
results of Chapter 7 and Chapter 10. 
A theorem of statistics can now be used to prove equation 
(12.57). If two independent random variables are distributed ac- 
cording to the distributions given by equations (12.55) and (12.56), 
then the sum of the two independent random variables is distributed 
according to equation (12.57). For a proof of this theorem, see 
Cramer [1946] (page 212). 
These equations hold for any net element anywhere in the 
and E 
#H,® plane. They also hold for E Thus the total 
Imax IImax° 
power is the sum of the power of the two systems. Also the power 
in any net element remains in that net element. It follows immed- 
iately then that equation (12.58) holds and that the integrals 
which represent 7, and 7,77 combine according to equation (12.59) 
where the integration over © may have to be from -7 to 7. Then 
from the definition given in equation (12.53), the desired pro- 
perties are proved. 
If equation (12.59) is approximated by a finite net, it will 
be seen that the equation is not a true identity for the finite net. 
The equation is valid only in the limit, and to prove the equation 
for a finite net, it would be necessary to consider a sub net ap- 
proaching infinitesimal areas inside of each net element. 
No theoretical analysis or finite net is capable of resolving 
the spectrum of 7 I+II into the spectrum of Nt and 737 if the power 
spectra overlap. However, if a swell power spectrum is added to a 
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