Periodicities when the Periods are unknown. 383 

 The factor for 2nd difference and sum series 



K>4.)HM'4.)H)H'-{-)H)H) 

 -K.)KH-7.)* 



Thus it can be shown that the factor for rth difference and sum series 

 will be (^-~ —^J - 



33. Now the last factors of all the terms in the scries (9-) contain a 



term of the type so that all these factors are positive, and, as 

 T 



before, it is easy to see that each of them has a tendency to vanish 

 as r increases, except in the cases where the terms of the type i 



have the value zero. The only terms of the type -t, whose values 



will be zero, are those which are formed from the sines of the angles 

 mb m 2mb n , 3mb n , &c, because these angles alone are some integral 

 multiple of 3tt. Thus when r is sufficiently great the series (9) will 

 be reduced to — 



Un sin (u n + P6») + u il2 sin (TT„ 2 + 2P5„) + u )is sin (TJ„ 3 + 3Pfr„) + , &c. 



If we curved now the first m values in order out of the last 

 horizontal line of our form which forms our rth difference and sum 

 series r it will be a variation with its full amplitude of the known 

 period m r and if the next m values be curved in the same way we shall 

 have a repetition of the same curve, and so on. 



34. Although we have shown that the factors of the form (l— 



V r 



when — is not zero, and r is sufficiently great, reduce to zero, yet the 



f 



reduction of the factors which are formed for the periods less than 

 mc is so much slower than the factors which are formed for the 

 periods greater than mc, that it is almost impracticable to continue 

 the process to the limit. We shall therefore in what follows, in- 

 vestigate a particular case of a known period, and we shall find it 

 advantageous to use partly our method and partly the usual method 

 of combining the series with the known period. 



35. Suppose we have a series of daily means of barometer observa- 

 tions in which we have detected a period (say) of exactly 36 days, 

 by the method already described. We shall, first of all, in the 

 following table, give factors of the kind in series (9) for different 

 values of r and different values of the periods of the type K. 



VOL. xxxvi. 2 D 



