u nr = 



V nr — T, 



«' nr = 



Vur — T( 



e = 



T t — T c 



e 2 = 





e 2 = 



X*' 



of the Earth? s Atmosphere. 289 



will be so nearly zero as to be regarded elusive and unimport- 

 ant. 



It may be remarked, in passing, that Professor NewcombV 

 paper does not sufficiently recognize these principles to do 

 justice to his criterion. Since this method may properly be 

 applied to many problems in meteorology, it should be fully 

 understood, and the following statement of it is made, slightly 

 changing the original notation. Let, 



n — the number of stations in each time-term. 



r = the number of time-terms (11-years, 3-years, year, etc.). 



v nr z= the departures from a long-record normal. 



r r = the mean departure in each term-time. 



r = the world-departure, or regional-departure. 



= the residuals. 



= the purely accidental local departures. 



= the mean accidental local regional departure. 



2 



— = the square of the mean v'. 

 n 



ne % = the mean square of v'. 

 n 



Hence, v = t -f- v', and r = r + e. 

 The original equation is by definition. 



v x + v 2 + v 3 + + V n = % n V = ?IT. (1) 



Square this equation term by term, form the squares and the 

 products, and take the sums, calling vv the successive pairs, 



2 n V + 2 $vv % = n*$T 2 . (2) 



If the variations v^ v 2 , t> 3 , etc. are purely accidental, there 

 will be as many positive as negative values of vv 19 and the 

 sum of the products will be zero, 



2 %vv 1 = 0. (3) 



Hence, if the departures are purely accidental, 



n 2 %r*-%y = = A = .Criterion. (4) 



If A = the value of the regional departure is zero ; if A 

 is a positive quantity, there is a true regional -departure or a 

 true world-departure ; if A is a negative quantity, one regional- 

 departure is hotter or colder at the expense of another regional- 

 departure. Substitute t -f v' for v, and t + e for r in (4), in 

 order to separate the purely accidental parts of the local and 

 regional departures. 



A = n 2 %{r + e) 2 - 2 n (r + v l )\ . Expand, (5) 

 A = ^S( To 2 + 2T e + e 2 )-2„(r 2 + 2ry + v' 2 ). (6) 



