Cycles of Rainfall 13 



To determine the value of a u multiply throughout 

 by cos kt and integrate between limits o and T. 



J*T pT /»T 



f (t) cos ktdt = A I cos kidt + o! j cos 2 ktdt 



o o o 



/T 

 sin kt cos kt dt + . . . 



o 



Or / / (0 cos kt dt = a x I cos 2 kt dt, since / cos kt dt and 



o o o 



/T 

 sin kt cos ta eft are both equal to zero and all the other 



o 



terms on the right-hand side of the equation, according 

 to our lemma, disappear. But 



C r ltu r 1 + cos 2 kt ,. . |\ sin2A'f| T 

 J cos*ktdt = J -^ dt-i\t + —-\- 



o o 



and as a result, we have 



/T 

 fit) 



a x — = I f (t) cos kt dt, or Qj = 2 - 



cos kt dt 



T 



Therefore ai is equal to twice the mean value of the 

 product /(0 cos kt. 



In a similar manner the value of any other coefficient 

 may be determined. Take, for example, b /r Multiply 

 throughout by sin nkt and integrate between o and T, 



C\ , x • , , , H  ,,., , n 1- cos 2 nkt ^ 

 j f (t) sin nkt dt = b n J sm« nkt dt = b n J ., - dt = 



bn {'I 2nk Jo J " 2 



/T 

 Therefore b n 



