Periodicities ivhen the Periods are unknown. 



381 



which, the original series is composed. The extent of this reduction 

 of range is shown in the following expression for the series that is 

 found after S repetitions of such smoothing operations : — 



Smoothed observation series* 



u= + fa sin (Uj + P5J } cos s A + {u 2 sin (U 2 + P6 2 ) } cos s h + , & c . 



2 2 



It will be seen at onca that as the period is smaller and smaller, we 

 have greater and greater reduction and vice versa. 



28. We shall, however, give a clue when a series contains enormous 

 irregularities, to the distribution of each number amongst the ad- 

 jacent columns, that is effected by our two methods of smoothing. 

 We shall give this by supposing that we have a series whose terms 

 are 13, of which each of the first and last 6 has the value zero, and 

 the 7th has the value 64. The result of six smoothing operations of 

 the second method gives 1, 6, 15, 20, 15, 6, 1. From this we see that 

 the reduction for the middle term which occupies the same column as 

 the number 64 in the original set is rapid, and the distribution 

 amongst the adjacent terms is smooth-flowing and of a convergent 

 character. 



29. We shall now apply our first method to a series whose terms 

 are 37, of which each of the first and last 18 has the value zero, and 

 the 19th has the value 343. Suppose we have a number 7 for a 

 working value of m to start the operations, we shall after three opera- 

 tions get a series of 19 terms whose values are +1, +3, +6, —11, 

 -27, -42, +91, +75, +57, +37, +57, +75, +91, -42, -27, 

 — 11, +6, +3, +1. From this we see also that the reduction is 

 rapid for the middle term, and the distribution amongst the adjacent 

 terms is in smooth-flowing waves which grow smaller and smaller and 

 ultimately vanish. 



Method of finding the Amplitude of a Variation of known Period. 



30. The processes which we are going to apply for this purpose are 



. nb 

 sin — . 



* This may be easily understood if we put n = 2 in the factor of the series 



. b 

 n sin — 

 2 



o • b b 

 . , 2 sin— cos — 

 sm o 2 2b 

 4 in paragraph 7. "When n = 2 the factor becomes 7= = cos -for 



2sin| 2sini 2 

 2 2 



the first operation, and in the same way it becomes cos— cos — , i.e., cos 2 — for the 



2 2 2 



second operation, thus for S operations it becomes cos 3 — . This may be taken as a 



2 



type of the factor for each expression of series of the type {« sin (U + P6)}. 



