14 Economic Cycles: Their Law and Cause 



is equal to twice the mean value of the product 

 f(t) sin nkt. 



Having found the algebraic values of the coefficients 

 in Fourier's series, we may now proceed to determine 

 their statistical equivalents in the case of annual rainfall. 



The Periodogram of Rainfall 



If the length of a cycle of rainfall were known before- 

 hand, the preceding exposition of Fourier's theorem 

 would suffice to determine, from the data of precipita- 

 tion, the amplitudes and phases of the harmonic con- 

 stituents of the Fourier series descriptive of the rainfall 

 cycle. But in the problem before us of analyzing the 

 rainfall data of the Ohio Valley, we do not know whether 

 there are many cycles or only one cycle or, indeed, 

 whether there are any cycles at all. And there is no 

 short method of solving the problem. 



Suppose, for example, it were assumed from a priori 

 considerations that the amount of rainfall is affected 

 by sunspots, and, as sunspots are known to occur in 

 periods of about eleven years, suppose it should be in- 

 ferred that the annual rainfall will likewise show a period 

 of eleven years. If the rainfall data of the Ohio Valley 

 are examined for an eleven years period, it will be found 

 that the data yield a definite amplitude and a definite 

 phase for a cycle of eleven years, but this fact is no 

 warrant for holding that there is a true rainfall period of 

 eleven years. Every other grouping of the seventy-two 

 years record will likewise show a definite amplitude 



