Cycles of Rainfall 15 



and a definite phase. The questions that one is in- 

 terested to have answered are: (1) What is the law of 

 the distribution of Fourier coefficients when the data 

 are analyzed for all possible periods; and (2) how may 

 the true cycles be separated from the accidental, 

 spurious cycles that are obtained when the data are 

 exhaustively analyzed? 



In Figure 3 the results of a detailed, laborious ex- 

 amination of the data of annual rainfall in the Ohio 

 Valley are presented in graphic form. On the axis of 

 abscissas are measured, within assigned limits, the 

 possible lengths of cycles in the 72 years of rainfall. 

 By extending the calculations to 36 years, we obtain 

 for the assumed periods a record of possible recur- 

 rences varying from 2, in case of the period of 36 

 years, to 24, in case of the period of 3 years. On the 

 axis of ordinates are measured the squares of the co- 

 efficients of the first harmonic in the Fourier series 

 corresponding to the lengths of periods recorded on 

 the axis of abscissas. The numerical values of these 

 squares are given in the fourth and eighth columns of 

 Table II in the Appendix to this chapter. The method 

 of deriving the values may be illustrated by taking the 

 cycle of 8 years. Suppose, as a first approximation, 

 that the equation to Fourier's series is put in the alge- 

 braic form 



y = F(t) = A + a x cos kt + 6 X sin kt = A + A l sin (kt + e). 



Then the corresponding arithmetical values derived 

 from the Ohio rainfall data are 



