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). 



z 



\ 



6C 



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 l cos kt + cuj cos 2 kt + a 3 cos 3 kt + . . . 

 + &! sin kt + 6 2 sin 2 kt + 6 8 sin 3 kt + . 



If in (1), we put, 



di = A l sin e v \ a^ = A 2 sin e^; a 3 = A 3 sin e 3 ; &c., 

 6 t = A l cos e^ b 2 - A 2 cos e 2 ; 6 3 = A J0 cos 6,; &c., 



We get, 



(2) y = A +A! sin (A;< -f c,) -h A z sin (2 W + 6 2 ) 



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. 



