10 



Economic Cycles: Their Law and Cause 



ured upon the line B C, the sinuous curve of Figure 2 

 is the graph of the function, y = a sin (nt+e). 



X p N Q /. J* 



Figure 2. 



The importance of simple harmonic functions in 

 the study of periodic phenomena grows out of the fact 

 that any periodic curve however complex l can be ex- 

 pressed mathematically by a series of simple harmonic 

 functions. By the help of Fourier's analysis a periodic 

 function may be put in the form 



(1) y =A + a y cos kt + a 2 cos 2 kt + a 3 cos 3 kt + . . . 

 + 6j sin kt + b 2 sin 2 kt + b s sin 3 kt + . . . 



If in (1), we put, 



a x = A x sin e x ; a 2 = A 2 sin e 2 ; a 3 = A 3 sin e 3 ; &c, 

 6 X = A l cos Cjj 6 2 = ^-2 cos e 2> ^3 = A z cos e 3 ; &c, 



We get, 



(2) y = A Q +A, sin (fcZ + ej + A 2 sin (2 Art + e 2 ) 

 + 4 3 sin (3Art + e 3 ) + . . . 



where y is expressed as a series of sines. In a similar 



manner, equation (1) may be expressed as a series of 



cosines, 



1 The few exceptions to the general rule are discussed in the 

 mathematical texts that develop Fourier's theorem. 



