252 General Strachey on the Correspondence of [May 31, 



gives a closer approximation to the actual observations than is got by 

 taking the simple arithmetical mean as the most probable value for any 

 year. 



In order to obtain a practical test of the probable physical reality of 

 the cycle of eleven years, I have calculated a series of mean values corre- 

 sponding to those given in Table I. for a series of cycles of five, six, seven, 

 eight, nine, ten, twelve, and fourteen years. I find that the mean differ- 

 ences between these means and the observed quantities, and therefore 

 corresponding to the mean differences shown in the last line of Table II., 

 are all within a very small fraction of one another and of the mean ob- 

 tained from the 11-year cycle — in short, that one cycle is in this respect 

 almost as good or as bad as another. 



The mean differences for the several cycles are given in the following 

 Table :— 



Table III. 



Cycles of 



Years of the cycle. 



Mean. 



1st. 



2nd.J 3rd. 



4th. 



5th. 



6th. 



7th. 



8th 



9th. 



flOth. 



11th. 



12th. 



13th. 



14th. 





5 years . . . 



in. 

 9-0 



in. 

 9-5 



in. 

 12-8 



in. 

 10-6 



in. 

 18-3 



in. 



in. 



in. 



in. 



in. 



in. 



in. 



in. 



in. 



in. 

 11-9 



6 years . . . 



7-7 



13-2 



11-5 



14-7 



13-2 



13-3 



















12-2 



7 years . . . 



15-9 



8-9 



6-7 



12-4 



8-5 



19-2 



12-5 

















12-1 



8 years ... 



5-2 



15-0 



12-1 



11-8 



8-0 



11-3 



14-9 



16-3 















11-8 



9 years . . . 



10"6 



11-2 



10-9 



12-6 



11-3 



16-7 



10-7 



13-2 



7-0 













11-6 



10 years ... 



8-5 



10-5 



7-8 



12-4 



17-7 



9-3 



7-9 



17-5 



8-4 



18-8 











11-7 



11 years ... 



11-5 



12-9 



6-0 



14-7 



13-1 



12-0 



12-3 



9-0 



11-4 



11-6 



7-6 









11-2 



12 years ... 



5-8 



11-4 



11-6 



13-0 



11-0 



12-7 



7-8 



15-4 



6-8 



13-4 



14-4 



139 







11-4 



14 years . . . 



17-7 



12-0 



4-5 



13-9 



7^5 



240 



14-1 14-2 



5-1 



6-3 



11 -3 



10-1 



12-2 



87 



11-9 



Now if in any series of quantities, such as the rainfall observations at 

 Madras, there be a law of periodicity, each observed quantity may be sup- 

 posed to be compounded of a periodical and a non-periodical element. If 

 we take the sum of a large number of cycles each of which coincides with 

 the cycle of periodicity, the non-periodical elements will tend to occur 

 in equal amount in excess and defect, and thus to be eliminated, and the 

 means for the successive years of the cycle, or whatever the intervals be 

 (which I will term cyclical means), will tend to indicate the periodical 

 elements for the successive intervals. At the same time, the differences 

 of these cyclical means from the several original quantities from which 

 they were obtained will approximate to the several non-periodical elements. 

 These differences I will call cyclical differences. 



In proportion as the periodical elements are small or large in relation 



