STUDIES IN STATISTICAL REPRESENTATION. 89 



from which, as before we obtain (18), (19), and (20), and 



further : — 



(20a)... b cos 2y = ^(y\ - t/' 2 +t/' 3 - y\) 



from which we have 



(20b)... cos 2y = fe/~ y ! a+y | 3 ~ y ^ sin /?. 

 (#1-2/2-2/3+2/4) 



7. Annual fluctuations: monthly data.— With monthly 



average results reduced to a uniform basis we put, [see (1) 



and (4)] 



(21) y{a +a x sin (^ + «i) + + a m sin m ( t T + a m )}^ 



= a x - a x cos (x + a x ) - — — a m cos m(x + a m ). 



m 



For the monthly averages, the integral is taken between 

 the limits to |w; |?r to fvr ; etc. 



Consequently if the right hand side of (21) be divided by J*r, 

 the monthly means (not aggregates) are obtained as a 

 series of twelve equations of which the first is 



6 I It 6 f )t 



Si = Cl - — 4 Cli \ COS (x + a } ) - 6 - — . \ a 2 \ COS 2(x + a 2 ) L L ... 



' I I. ' ■( lo 



the second is 



S 2 = a - — • di 1 cos (a + ai) L - — . i a 2 J cos 2(as + a 2 ) !• — ... 



7T 7T i 



and so on. 



Summing these vertically, we see that 



Si + s 2 + + S12 = 12 a ; thus 



(22) o* — T2" ^1 Sk 



It is desirable for the purposes of computation to form 

 the quantities Si - a ; s 2 - a ; etc., which will be denoted 

 by fi, r 2 , etc. Thus 



f 1 = — [^{COS a 1 - COS (<h + 30°)} + ^ a 2 {cOS 2a 2 



-cos 2 0* 2 +30 o )} + ■§ a 3 {cos 3a 3 -COS 3 (03+30°)} + ...] 

 r 2 =— [oi{cos (ai+30°)- cos K+60°} + ■|a 2 {cos2(a 2 +30 o ) 



7T 



and so on. —cos 2 (a, + 60°)} + ...] 



