A STUDY OF CORRELATIONS AMONG TERRKSTRIAL TEMPERATURES. 323 



Also, by squaring the equation (1) and adding the r squared equations we find 



rr I'.z- = l"-..v- + 21' vv' 

 where Siv/ represents the sum of the rn{n — 1) products of each two departures in 

 every term. If these departures (; are purely accidental deviations from means the 

 ratio of %vv' to Xv"^ will tend toward zero as the number of terms is indefinitely increased. 



Dropping them we find the condition 



Hence, if we put, 



n-ZjT- = z,jV 



A = /rS,T^' - S,,y (4) 



the criterion for the independence of each /' from the others will be 



A = (5) 



If this equation is not satisfied within the probable limits of the accumulated 

 accidental errors, it will show that the hypothesis of the complete independence of the 

 temperatures of the different regions is not established, and that there is some corre- 

 lation between them. This may arise from any common cause affecting the tempera- 

 ture at two or more of the stations. Let us suppose a varying cosmical cause affecting 

 the entire earth, the result of which is to raise the world-temperature during any one 

 term by an amount r,,. Each observed departure will then be made up of two 

 parts : — 



(1), the common departure To for the whole world ; 



(2), an accidental local deviation peculiar to the region. We shall then have, as 

 the value of each individual departure in any region during any one term 



»,■ = T„ 4- v/ (6) 



v' being the purely accidental deviation, whose mean value is e. 



Form the sum of the squares of the equations (6) for the n values of the Vi for any 



one term 



S.t,.2= „T--^ 2Sv.' + X.v'- (7) 



The mean value of?/ being the same as that of e, and each value of // being indepen- 

 dent of To, we have the probable equation 



Summing the equation (7) for the /• time-terms and putting e for the mean r'- we 



have 



S,. /•'' = n'ljTl + nre^ (8) 



