STUDIES IN STATISTICAL REPRESENTATION. 101 



and the general lag will be 

 (57)... ^o = a o — a V 



This expression which is empirical, however, loses 

 significance as the similarity of the two curves disappears, 

 and in any case can be regarded only as of the nature of a 

 rough indication of the general time relation between them. 



14. Analysis of sine and cosine curves;— When a curve 

 is of the type of a simple sine curve, differing only in respect 

 of absence of symmetry, it may be represented by 

 (57a)... i/ = A + Bsin^ + C cos x 

 which is identical with 

 (57b) y = A + b sin (x + (3) + c cos (x + 7) 

 since it is easily shewn that 

 (58)... B = b cos |8 — c sin y; and 

 (58a)... C = b sin <3 + c cos y. 



In all cases we have 



(59)... A = l^t/ k 

 n 



If we put as before r k = y k - A, we shall have for quadri- 

 mestral division 



(60)... B = ^^;C=-||; 



for quarterly division 



(61)... B= g"(n+ fa-rs-r^,'; C — ^(r 1 —r 2 -r 8 + r 4 ) 



and for division into five parts 



the coefficients of which may also be written 

 w/{5(l + cos 36°)} and 4tt/(5 sin 36°). 

 Where the form in (57a) satisfactorily represents the data, 

 it is occasionally necessary to determine the point where 

 the curve has a maximum or minimum value or equals the 

 average value (A). 



The points where y = A are given by 

 B sin 6 + C cos 6 = 0. 



