138 Economic Cycles: Their Law and Cause 



origin in a natural cause and persisting with a change in 

 the samples of statistics, and a spurious periodicity 

 which is purely formal, having its origin in accidental 

 characteristics of the statistical sample and disappear- 

 ing, or radically altering its character, when different 

 samples of statistics are made the basis of the computa- 

 tion. (3) The method must not only make possible 

 the isolation of real periodicities, but it must likewise 

 enable one to determine their essential characteristics, 

 their length, phases and amplitudes. The method we 

 adopted in our researches, which is based upon the 

 harmonic analysis, satisfies these three conditions. 



The result of our investigation as to the periodicity of 

 rainfall in the upper Mississippi Valley was the dis- 

 covery that the annual rainfall passes through two 

 cycles of approximately thirty-three years and eight 

 years in length. The amplitude and phases of these 

 two cycles were ascertained, and the equations to the 

 separate cycles were calculated. The two cycles were 

 then superposed, thus giving the general cyclical move- 

 ment of rainfall; the equation of this compound cycle 

 was computed and the graph was drawn. It was found 

 that the curve of the rhythmical movement of rainfall 

 computed from the equation to the superposed cycles 

 fitted excellently well the actual observations of rainfall. 

 These results constitute the solution of the first part of 

 our general problem : Rainfall in the principal crop area 

 of the United States passes through cycles of thirty- 

 three years and of eight years. 



The caution that should be observed in the use of our 



