Cycles of Rainfall 11 



(3) y =A O + B! cos (kt-eJ + B^ cos (2kt-e 2 ) 

 + B 3 cos (3 ta- 3 ) + . . . 



In the use of Fourier's theorem for the purpose of 

 analyzing periodic phenomena, we follow a process 

 analogous to the use of Taylor's theorem in the simpler 

 demonstrations of mathematical economics. By far 

 the greater part of Cournot's pioneer treatise and of 

 subsequent work of his school is based upon the as- 

 sumption that, if the economic function under investi- 

 gation is y=f(x), then f(x+h) may be expanded by 

 Taylor's theorem, and the first terms of the series may 

 be used as an approximation to the form oif(x). Simi- 

 larly, in our use of Fourier's series, the attention will be 

 focussed upon a few harmonics as a first approximation 

 to the solution of the problem in hand of expressing 

 in mathematical form the periodicity of annual rainfall. 



Assuming that any periodic function may be ex- 

 pressed as a Fourier series, the problem is presented of 

 determining the values of the coefficients. The series, 

 as we know, is of the form 



y = f (t) = A + aj cos kt + 0% cos 2 kt + .. 

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



What are the values of the first term and of the co- 

 efficients of the sines and cosines? In order to deduce 

 the necessary values, we shall have need of the follow- 

 ing lemma: 



If m and n are two unequal integers and k is put equal 

 to --, then 



