near am/5, the second value of m would show this more clearly. 
The power integrals under study in this paper are extremely 
complicated functions. Their analysis is consequently also ex- 
tremely complicated. The numerical methods presented by Tukey 
[1949] and Tukey and Hamming [1949] are the only methods of analy- 
sis which permit as a final result a correct estimate of how 
accurate the calculated power spectrum is.* 
Associated with the final m numbers obtained in the analysis 
is a value, f, which is called the number of degrees of freedom 
of the value of UL: The value of f can be computed fon equa=- 
tion (10.39). The larger the value of f, the more reliable the 
power estimates of the spectrum. Equation (10.39) shows that 
the larger the value of N the larger the value of f and that the 
larger the value of m, the smaller the value of f. Thus greater 
resolution, which requires a large m, sacrifices accuracy of 
analysis unless a very large N is chosen. 
The number of degrees of freedom of the sample, f, can be 
used to determine the reliability of the power spectrum estimates. 
Tukey and Hamming [1949] have shown that the values of Un given 
by equation (10.32) are distributed according to a X * distri- 
pution with f degrees of freedom. Table 16 gives the important 
numbers connected with this distribution. The first values are 
expressed in decibels for the convenience of electronic engineers 
*This statement is made with the knowledge that many incorrect 
and inadequate methods for determining hidden periodicities and 
their significance are to be found in current geophysical 
studies. A prime example of how to do things wrong is found 
in the current claims of Langmuir [1950]. 
- 201 - 
