STUDIES IN STATISTICAL REPRESENTATION. 77 



(1) y == a J- Oi sin (# + aj 4- a 2 sin 2 (# + « 2 ) + ... 



+ a m sin m (# + a m )... 

 the complete period being denoted by x = 2tt. Or again, 

 they may be represented by a similar series containing 

 instead cosine terms, viz. bi cos (x + ft), b 2 cos 2 (# + ft), 

 etc.; or yet again, if the series contain both sine and cosine 

 terms the angles a lf ft; a 2 , ft, etc., may be omitted, the 

 general term being a m sin mx + b m cos m#, in all cases m 

 being an integer. 



In the above formula 04, a 2 ...a m ... represent the intervals 

 between the origins of particular elements [the complete 



. , ;:' . , . , '. , 2tt 2tt 2tt 2 tt -, 



periods of which are respectively — -, -— , -— , ... — ... | 



1 Z 6 m 



and the beginning of the period under review, say the com- 

 mencement of the year or month, or of the day, as the case 

 may be. They may be called " epochal angles." The 

 factors »i, a 2 ...»m... are the semi-amplitudes of the oscilla- 

 tion. Epochal angles are not necessary however when the 

 series contains both sines and cosines. It may be noted 



that the periods -i -^, ... — being all sub-multiples of 

 2 3 m 



2k, are exactly repeated with tlie larger period 2tt. 



By way of illustration it may be said that the secular 

 means of weekly, monthly, or quarterly variations of tem- 

 perature, of rainfall, etc., may be replaced by a curve of the 

 form indicated in (1), in such a way that the area standing 

 on the abscissa representing any interval of time and 

 bounded by the curve, the axis of abscissae, and the two 

 limiting ordinates, will give the same average result over 

 the interval of time as was originally furnished by the data. 



In many cases, however, statistical results may disclose 

 oscillatory fluctuations of which the absolute amplitude 

 increases as some function of the time : hence in dealing 

 with certain classes of statistical data it may be necessary 

 to write instead of equation (1) 



