48 Generating Economic Cycles 



ness of this assumption. An ample description of these 

 graphs will be given later on. 



2. The authority of Professor Schuster is against the 

 early elimination of the secular trend before the Fourier 

 terms are computed: ''Very considerable labor has 

 sometimes been spent in eliminating secular variations 

 and other known periodicities before the hidden 

 periodicities are searched for. We may reasonably 

 ask the question, what object is thereby gained? It is 

 one of the great advantages of Fourier's analysis that 

 each of its terms is independent of the others; and if 

 we wish to determine any particular coefficient it is 

 imnecessary to begin by eliminatmg the others. The 

 analysis itself performs that process in the best possible 

 way, if the coefficients are obtained by arithmetical 

 calculations. . . . The general rule may be given, that 

 it is the best to eliminate as few variations as possible, 

 and to carry out the elimination at as late a stage of the 

 computation as possible." ^ 



For these reasons we have computed the Fourier 

 terms directly from the index numbers in the raw state. 

 The results of the computation are given in Table II 

 of the Appendix. The headings of the table will be 

 understood from an examination of Fourier's series 

 when it is expressed in the following two forms: 



(1) y = Ao -\- ai cos kt-\-a2 cos 2kt-i- . . . 

 + bi sin kt + &2 sin 2kt + • • . 

 {2)y = Ao + Ai sin {kt + eO + A^ sin i2kt + 62) + • . . 



The amplitudes of the terms in (2) — the ^-coeffi- 

 cients — are obtained from the corresponding coeffi- 



1 "On the Investigation of Hidden Periodicities," p. 34. Cf. also p. 38. 



