A STUDY OK CORRELATIONS AMONIJ TERRESTRIAL TEMPERATURES. 325 



the simultaneous temperatures in the different regions being independent, or affected 

 b}^ a common cosmical cause, one region on the average becomes hotter or colder at 

 the expense of another. In other words the conclusion would be that when the tem- 

 perature was above the normal in one region, it was more likely than not to be below 

 it in other regions, and vice versa. Thus the conclusion as to a positive correlation, — 

 no correlation or a negative correlation — depends upon whether A is positive, evanes- 

 cent or negative. 



§ 4. Case when Different ]VeigJits are Assigned to Different Regions. 



For the sake of simplicity we have developed the preceding method on the 

 assumption that in determining the general departure r the different stations or regions 

 are all entitled to the same weight. But if the accidental deviations are smaller at 

 some stations than at others it is clear that the observations at such stations will be of 

 greater weight for the detection of cosmical causes. We should therefore assign weights 

 to the several stations determined b}' the usual methods. Let these weights be 



Wi, "'z. • • •> w,. (11)' 



and let us call IF their sum. The preceding equations will then be replaced by the 

 following : 



Instead of using (1) for determining r we use the equation 



If T = u\i\ + WA\ +•••+;(•() = 1,w:c. n 2) 



1 1 ' 2 2 ' * n n lit \ / 



Let us put e, for the mean accidental deviation of v^ and e^ for that of r. The 

 mean deviation of any one product w^Vi is theii ?(^j€, and the squared mean deviation of 

 the sum of all these products for any one term, if uncorrelated, is 



1 t 



The mean €^ should in this case satisfy the equation . 



W\' = w;e; + «,'/e/ + • • • + wX' (15) 



If the observed deviations v are wholly in the nature of accidental deviations from a 

 mean value, we may take for each £;- the mean of all the v^ ; and r being then a purely 

 accidental deviation of the mean, we should have the probable equation 



ej = mean t^ 

 The criterion for deciding whether the deviations are purely accidental may therefore 



A. P. S.— XXI. SS. 13, 1, '08. 



