Generating Cycles in a Century of Prices 49 



cients in (1) by means of the foraiula A = \/ d'+h'-. 

 Aq is equal to the mean value of the 96 index numbers. 

 The lengths of the periods in the second colunm of 

 Table II are obtained by di\'iding 96, which is the 

 number of years of obser\'ations recorded in the Sauer- 

 beck index numbers, by the consecutive integers that 

 are given in the first column. The constant k in the 

 fomiulae (1) and (2), which does not appear in Table II, 



oprvo 



is equal to -^ = 3° 45'. The constants e in (2) do 



not occur in Table II, but their values may be obtained 

 from the corresponding values of a and h in (1) by the 



relation tan e = t. 

 6 



We shall now consider the conclusions that may be 

 drawn from the data of Table II and we shall begin with 

 the last column which gives the values of A-. If we let 

 the eye run down the last colmnn, it will note that at 

 four places the values of A ^ assume special importance — 

 for the periods of 96 years, 48 years, 24 to 16 years, and 

 8.7 to 7.4 years. A moment ago when the question of 

 eliminating the secular trend was under consideration, 

 the decision was reached to permit the early terms of 

 the Fourier series to pro\dde for the secular trend rather 

 than to adjust the raw figures of the observation ac- 

 cording to a more or less arbitrary assumption as to 

 the type of curve which might be appropriate to repre- 

 sent the trend. It now seems reasonable to conclude 

 that the large values of yl^ at 96 years and 48 years, and 

 possibly those between 24 and 16 years, are caused by 

 the general trend of the figures. Inasmuch as the first 

 of these covers the whole range of the observations and 



