78 G. H. KNIBB8. 



(2) y' = f(x){a -\-a l sin (#+<*,) + ... +a m sin («+%)+..«} 



that is to say, the values of the fluctuating or periodic terms 

 have to be multiplied by the factor f (#), or in other words 



(3) y = y'lf(x). 



For example, with a growing population, the changes in 

 the number of births from month to month shew a seasonal 

 or periodic fluctuation, the amplitude of which however is 

 itself growing in proportion to the total population : while 

 the mean monthly rainfall of a country which was becoming 

 arid, would similarly shew a periodic fluctuation, the ampli- 

 tude of which was diminishing in the ratio of the diminution 

 of aggregate annual rainfall. This method will always be 

 satisfactory when the amplitude of the fluctuation is rela- 

 tively small as compared with ?/, in other words, in all cases 

 where the amplitudes vary only as f(x) in (3), and are 

 or may be considered to be, unaffected by the magnitude 

 of the periodic term itself. 



With regard to the form in which the problem may be 

 actually presented it may be noted that data may be either 

 in the form of instantaneous values — that is, values of the 

 function with different values of x as argument, — or group 

 values, viz., totals or averages between certain values of x. 

 The latter form is that which usually occurs in statistical 

 applications. Instantaneous values are of course directly 

 represented by (2). For group values we have, on integrat- 

 ing (1), 

 (4) Y=fydx = a x - a x cos (x + a i) - \ a 2 cos 2 (x + <* 2 ) - 



... - — a m cos (x + O... 

 m 



Y denoting an area. Hence group values are essentially 

 the values of this integral between the limits x Q to x x ; x 1 to 

 x 2 ;...x n —i to x n ; n denoting the number of groups. In most 

 instances practically occurring, the data do not furnish Y 

 but Yjf&x, i.e. Y/(x 1 - x ); Y/(x 2 -x 1 ); Y/(x 3 -x 2 ); etc., for 

 the different limits. 



