260 General Strachey on the Correspondence of [May 31, 



nitely prolonged series of observed quantities A 1? A 2 , . . . from their mean 

 M a , then the probable value of e a deduced from the mean of (m) cycles, or 

 series of observations of A, also irrespective of sign, will (according to 



E a 



the known laws of the combination of errors) equal -r=. The same 



will hold good of all the series B, C, &c. At the same time, as there is 

 no periodicity, and all the observations are presumably liable to errors or 

 irregularities of the same general character in a positive or negative di- 

 rection, the quantities E B , E 4 , &c. will, in a sufficiently prolonged series, 

 tend to equality one with another and also with the mean deviation, 

 irrespective of sign, of all the observed quantities from the arithmetical 

 mean of the whole of them, which call E. 



Hence, when there is no periodicity, the sum S, as before defined, tends 



to become n x -7=. 



v m 



We are thus led to the conclusion that the consideration of the succes- 

 sive values of the quantity S as the number of cycles of periodicity 

 increases affords a true criterion of the presence or absence of a periodical 

 element. If as (m) increases this quantity is gradually reduced in a ratio 

 E 



approximating to we may infer that the periodicity is small or does 



not exist. On the other hand, if the value of S tends to become invari- 

 able, and continues to be of considerable numerical magnitude after a 

 prolonged series of cycles, the existence of a true periodical element is 

 apparent. 



In the Madras observations the successive values of — (obtained by 



combining one after another the observations of one cycle of 11 years, of 

 two such cycles, of three, and so on, till the whole six are united) are 

 shown below, contrasted with the corresponding calculated values 

 E 



of V'm Table XII. 



Number of 

 C}-cles. 



From observation, 

 8 



n 



Calculated, 

 E 





13-2 



]32 





9-5 



100 



3 , 



73; 



6-9 



4 „ 



4-9 



04 





4-5 



5-3 



6 



4-5 ? 



51 



18 „ 



3-4 

 32 

 2-7 



3-8 

 30 

 2-7 



