1877.] 



the Rainfall with the Sun-spot Period. 



259 



corresponding to the sun-spot period, but to point out in the case of the 

 rainfall not only has no such correspondence been established, but that 

 there has been no sufficient evidence adduced of any periodicity at all. 



Appendix. — Received May 17, 1877. 



Suppose the numerical values of such a series of observations as that 

 discussed in the preceding paper to be represented by 



A x , A 2 , A 3 , ... for the first year of each cycle of (n) years, 

 B l5 B 2 , B 3 , ... for the second years, 

 and so on for (m) cycles. 

 Also suppose M to represent the arithmetical mean of all the observa- 

 tions, and j? a , p b , p c , . . . to be the periodical variations from the mean, 

 M, for the several series of the A, B, . . . years of the cycles, and the 

 corresponding non-periodical variations to be a v a 2 , a 3 , . . . , b v 6 2 , 5 3 , . . . , 

 and so forth — all these quantities being affected by their proper signs. 



Then if M a represents the mean value of any series A v A 2 , A 3 , . . . , 

 we should have for (m) cycles — 



where e a with its proper sign represents the non-periodical portion of the 

 mean value of the A series for (m) years. 



With a sufficiently prolonged series of cycles the quantities a v a 2 , . . . 

 will tend to cancel one another and e a will disappear ; so that M a will 

 then become equal to M+p a , and M a — M=p a . 



If, therefore, there is a truly periodical element, the difference of the 

 mean of all the observations, and of the mean of any series (A) of the 

 cycle of periodicity, will (in a sufficiently extended series of observations) 

 tend to be identicai with the periodical element (p a ) for that cycle. 



This holds good of all the series A, B, C, . . . ; and therefore the sum 

 of all the differences last mentioned for the several series will tend to 

 equality- with the sum of all the periodical elements of the several series. 

 This will be true whether we regard the algebraic sign or not. If we so 

 regard it, the sum of the differences will evidently become equal to zero, 

 as also will the sum of the periodical elements. If we disregard the 

 signs, the sums will have a numerical value, which call S. This, if 

 divided by the number of years in the cycle of periodicity, will indi- 

 cate the mean deviation, either in excess or defect, of the periodical ele- 

 ments from the arithmetical mean of all the observations. 



Hence, when there is a true periodicity, the sum of these differences 

 taken without regard to sign tends to become an invariable quantity, and 

 the numerical magnitude of this quantity indicates the magnitude of the 

 periodical variation. 



Let us next consider the case in which there is no truly periodical 

 element. Here M a =M + e a and M a — M a=e a . If E a represents the mean 

 numerical value, irrespective of sign, of all the deviations of an indefi- 



T 2 



