12 Economic Cycles: Their Law and Cause 



cos mkt cos nkt dt = 0, 

 sin mkt sin nkt dt = 0, 



r 



o 



J 1 



o 



/T 

 sin raA;£ cos nkt dt = 0. 



The lemma may be proved to be true by evaluating the 

 three integrals according to the usual methods. The 

 first integral, for example, becomes 



/t pr 



cos mkt cos nkt dt=\ / { cos (m—n) kt+cos (m+n) kt\dt 



o o 



rsin (m-n) kt sin (m + n) kt~\ T 

 " [ 2 (m-n) k + 2 (m + n) k \ 



But k = -7=- , and, consequently, I cos mH 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 /»t pi pr 



f (t) dt = A j dt + Oj / cos kt dt + h J sin ktdt + . . . 



o o o o 



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

 side of the equation will vanish, and consequently 



/T 

 fit) 



j \f(t)dt=A Q I 'dt = A T, or A 



. dt 



o 



/'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. 



