Periodicities when the Periods are unknown. 



377 



less than each of K 2 , K 3 , &c, then, as before, we can show that q 1 is 

 negative, and each of q 2 , q 2 , &c, is positive. 



Let <7i= — Qu then the factor — -\ =( 1— IV will be equal to 

 ^l + -ty = ^S^tiy, and the value of each of the factors ^> 

 (& — , &c, where r is sufficiently great, reduces to zero. Thus, 



V 23 / 



when mc is greater than the period K l5 less than K 2 , and less than 

 double the period of K 1? and the number of operations of taking 

 successive means and remainders are sufficiently performed, the re- 

 maining series reduces to {u^ sin (Vx + Pfrj)} ^~^q~^ ' or a su ^" 



ordinate series in the observation series whose period is K T , and the 

 value of each term of the series is greater than the corresponding 



term of the original series : since ^QllLi^ is greater than one, and 



when r is large enough, f ^ 1 jl ^ \ can be made as large as we please. 

 V Q,i / 



20. Up to this point we have treated the above subject by sup- 

 posing that m is an odd number. By simple inspection it will be 

 seen that when m is an even number the results will be qnite similar 

 to the one when m is an odd number, but the labour of operations of 

 taking successive means will be twofold. Because the mean of m 

 values from which the mean is taken will not correspond to any of 

 the values in the lot, but will correspond to the middle of some two 

 consecutive values, and for this reason the mean should be written on 

 the line instead of in the column on our form ; and after the opera- 

 tions of these first means are performed, we must take the mean of 

 two consecutive means found out first, at both sides of any particular 

 column, and put it in the column. Then these second means thus 

 found are to be subtracted from the corresponding values from which 

 the two sets of means are derived. Similarly from these remainders 

 new means and new remainders are to be found precisely in the same 

 manner as spoken of above. In order to keep our language the same 

 whether m is an odd or an even number, we shall always, in the case 

 where m is an even number, give the name of first set of means, 

 second set of means, &c, to those means found by two of the opera- 

 tions described above. 



21. This suggests that we must always prefer m as an odd number, 

 except when there is real need of preferring m as an even number, 

 since the labour of operations, when m is an even number, is greater 

 than when m is an odd number. It will be seen further on that 

 there is sometimes real need of taking m as an even number. 



