Periodicities when the Periods are unknown. 



379 



simple harmonic period of exact duration — 25 days — that we have 

 been suspecting and searching for, the process we have been per- 

 forming will still tell us if there be any simple harmonic period of 

 any duration between (say) 20 and 30 days. The only difference 

 we shall have is in these cases that we shall have to increase 

 the number of operations to find the remaining series. In the fol- 

 lowing table we shall give a few numerical results of the factors — 



35tt 

 20 



35 sin -_- 



20 J 



sin 



1 — 



35^ 

 25 



35 sin — 

 25J 



1-. 



35tt 

 30 



35 sin 



30 J 



When r = l . . 



„ r = 7 .. 



„ r-U.. 



„ r = 12. . 



„ r = 16.. 



„ r = 18.. 



„ r=24. . 



„ r=36.. 



Numerical values of the factors. 



35tt 

 20' 



35 sin 



20 



•1291 



•339 



•802 



•293 



•977 



•895 



•43 



•12 



35tt1 

 25 I 



1- 



35 sin — I 

 25 J 



1 



3 

 8 

 10 

 22 

 32 

 105 

 1076 



•2168 



•886 



•441 



•25 



•26 



•81 



•0 







35?r 

 30 



35 sin 



30 



1 



2 

 4 

 4 

 7 

 10 

 21 

 100 



•1367 



•452 



•094 



•654 



•770 



•04 



•66 



•7 



The object of introducing the above table is only to give a rough 

 idea to the reader of the rapidity with which the numerical values 

 of these and such similar factors increase after a certain number of 

 operations of the remaining series. 



23. Suppose we have suspected only the period of 25 days, but in 

 reality there is not any period between 5 and 35 days ; the result will 

 be, as investigated before, that the range of remaining series will be 

 lessened and lessened, until each term of the last remaining series 

 will be less than two in the last place of the figures. 



24. Suppose in the daily mean series we have a real existence of 



the two periods (say) 25 and 20 days, we shall still find the existence 



of the period of 25 days. Because the numerical value (regardless of 



. 35tt 



sin . 



25 



sign) of the factor is greater than the numerical value 



35 sin ^ 



