A STUDY OF rOHRELATIONS AMONG TERRESTRIAL TEMPERATURES. 327 



We may, if we choose^ reduce the results for any number of regions in the same 



way by taking the regions in pairs. By squaring (1) we have, for any one term of 



observation, 



nV=2,r7+2S,,»vv (19) 



in which each individual product tw. is formed from each pair of the individual v's 



for the time-term, so that we have n(n — 1) products v,:?v- for each of the r time-terms. 



Summing the series for all the time-terms during which w remains the same, we 



have 



n-2/ = S,,,.r+22,.,.,?y), (20) 



Combining this with (9) we have 



«(i<-l)V5 = 2S,,.,r,t., 



Taking tq to represent the mean value of the cosmical fluctuation through r terms, we 

 have 



Also, 



S.T.r = rV (21) 



where, for brevity, we put t^ for the triple summation of the products. We are thus 

 enabled, when we so desire, to compute A, and hence the value of V, for each time-term 

 and each pair of stations taken separately. The final mean of t/ which we thus 



derive instead of (10) is 



2'2hiv' 



Mean t/ = -. -r (22) 



" n(>i — 1) 



The number of combinations of 71 stations being [«(h — l)]/2, this is equivalent to 



Mean t/ = Mean vv' (23) 



which may be found by summing (18) for the pairs of stations and all the time-terms. 

 For considerable values of n this equation is more laborious in use than (10) or (17), 

 but it has the advantage of showing whether a correlation among the departures of 

 temperature exists for all the stations, or is confined to a limited number of stations. 

 The preceding value of v has been derived for the sake of simplicity, as if the 

 weights were all equal. When the pairs of stations are all considei-ed individually, no 

 difference of assigned weights will affect the resulting individual value of V- But if 

 we combine the [//(n — l)]/2 individual values thus derived, we must assign them 

 their proper weights. These we find by dealing with (IG) in the same way that we 

 have dealt with n^T- when the weights were each 1. By squaring (12) and summing 

 for the r time-terms, we find 



s,s,«,>'r,^ = s, Wh' - 2S, ,, ,«^.«v^•% (24) 



