Cycles of Rainfall 13 



To determine the value of a b multiply throughout 

 by cos kt and integrate between limits o and T. 



/T /T /T 



f (t) cosktdt =A I cosktdt + at I cos*ktdt 



O 00 



+ b ! / sin kt cos kt dt -\- . . . 



o 



Or I f (t) cos Atf eft = a x I cos 2 Atf eft, since I cos Atf eft and 



O 00 



/ sin kt cos kt dt are both equal to zero and all the other 



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

 to our lemma, disappear. But 



r* f T 1 + cos 2 kt , , r sin 2 ktl T T 

 I cos*ktdt= I ^ dt = i [^ + 2fe J Q = -^ 



o o 



and as a result, we have ^T 



/ / (0 cos kt dt 



T C T * 

 fll - = I / (t) cos Atf d<, or a x = 2 - ^ 



o 



Therefore a\ is equal to twice the mean value of the 

 product /(O cos kt. 



In a similar manner the value of any other coefficient 

 may be determined. Take, for example, b n . Multiply 

 throughout by sin nkt and integrate between o and T, 



f*t r T . r r l-cos 2 nAtf ,. 



/ / (0 sin nkt dt = b n I sin* nkt dt = b n I ^ -dt = 





/T 

 . Therefore b n 



