12 Economic Cycles: Their Law and Cause 



I cos mkt cos nkt dt = 0, 



o 



I sin mkt sin nkt dt = 0, 



o 



/sin rata cos nM d = 0. 



o 



The lemma may be proved to be true by evaluating the 

 three integrals according to the usual methods. The 

 first integral, for example, becomes 



fcos mkt cos nkt dt = | I \ cos (m-n) kt+cos (m+ri) kt\dt 

 J o 



[s\n(m-n)kt sin (m + ri) kt~\ T 

 2 (m-n) A; 2 (m + n) k \ 



2ir /* T 



But k = - , and, consequently, / cos rafa cos nkt dt = 0. 



With the aid of this lemma we may proceed to evalu- 

 ate the coefficients in Fourier's series. If we integrate 

 the series between the limits o and T, we get, 



t = A f^ + d! C T 



But all of the terms except the first on the right-hand 

 side of the equation will vanish, and consequently 



f V f(t)dt 



/T 



Since / f(t)dt is the area of the original curve for one 



o 



whole period T, the constant term in Fourier's series is 

 equal to the value of the mean ordinate of the original 

 curve. 



