380 Mr. Y. N. Nene. On a Method of Tracing 



357T 



(regardless of sign) of the factor sin (refer to the cnrve of 



35 sin — 

 20 



factors) and consequently ( ^~^ - 1 can be made as large as we please 

 in comparison with ^l^Ll^ when r is large enough, where ^— 



stands for the former factor and ( — ^\ stands for the latter factor. 



Thus if the compound result in the last remaining series be curved 

 out on a curve form the maxima and minima of 25 days' period will 

 be apparent. Similarly if any number of compound periods be mixed 

 in the last remaining series, still by sufficiently increasing the number 

 of operations the 25 days' period will be apparent. In conclusion we 

 can say that among the periods which may have mixed in the last 



remaining series the one of which the factor of the type ^-^ has the 



greatest numerical value (regardless of sign) will be apparent, 

 whether the period is of 25 days or not. 



25. The only case in which the period will not be rendered dis- 

 tinctly apparent is, that when the two periods are such that their 

 factors of the first mean series are equal to one another, and then the 

 two variations will continue combined in the same proportions as in 

 the original observations. 



26. We have in the above example stated that when we find syste- 

 matic movements in the daily mean curve, of which the duration is 

 from 5 to 10 days, we must first work the series with five day 

 operations, &c. It is not, however, essential, when the period which 

 we are seeking is of a considerable range compared with that of the 

 shorter period, then the period can be found without it ; but when 

 the range is very small the first process of smoothing is quite 

 essential. It is not also essential that we should stop the operations 

 when the limit of the mean series is reduced to less than 2 in the 

 last place of figures in the series, but we may stop at any stage of the 

 process after considering what reduction will take place in the range 

 of the period we are seeking. This is merely a process for making 

 the series of smooth- flowing numbers without sensibly reducing the 

 range of the larger periods. 



27. The process of smoothing series generally in practice is to take 

 the mean of two consecutive numbers, and put it down below and on 

 the line between the two numbers, and so on with each pair of 

 numbers in succession. Next from this new series the same thing is 

 repeated up to a certain number of operations. One effect of this 

 process is to reduce the range of each simple harmonic variation of 



