290 F. II Bigelow — Studies on the General Circulation 



The terms 22t <? and TL%t v' are each equal to zero, and since 

 ne 2 =*2 n v /2 by definition, 



A ±= n 2 T * — nr 2 = n 2 %T 2 — \v 2 = 0, for accidental v n . (7) 

 A = n(n—l)r a = w'St' — \v\ for n stations. (8) 



Similarly, 



A r = rn(n — l)r 2 = n 2 %r 2 — 2 nr ^ 2 = 0, for r series. (9) 

 This is Newcomr/s Criterion for accidental departures for 

 a large regional. or world departure t . (1) Take the mean 

 of the departures v n for the t ; square them and take the sum 

 multiplied by n 2 ; (2) square all the v in the ^-stations and 

 ^-time-terms and take the sum ; (3) divide the difference by 

 rn(n—l) to find t , the mean regional departure. These should 

 be completed by consecutive series in succession, as 1872-76 

 for t at 1874, 1873-77 for t at 1875, and so on, thus elimina- 

 ting all periods shorter than r time-terms, 5 years in this case. 

 The probable error of the world or regional departure from 

 the normal is found as follows : The probable error is, by the 

 theory of least squares, 



u* = v 2 - t\ (10) 



Substitute v = r + v' and t = t„ + e, 



ii* — v F2 -e 2 - 2t (v'+ e). (11) 



In the summation the product 22t (v'H- e) = 



\u 2 = Xv' 2 - ne 2 . (12) 



Substitute 2 n i> /2 = ne* and ne 2 = e 2 , 



% a u 2 = ne 2 -e 2 = (n - l)e 2 = Xv 2 - nr 2 . (13) 



Similarly for a series of r time-terms, 



r(n - l)e r 2 = X r v 2 - n\r 2 , (14) 



from which the value of € or e r are found. 



These mathematical principles are valuable in some cases of 

 pure homogeneous data, simple definite period, no inversion 

 effects and no complex absorption and radiation effects of land 

 and water masses, but they must be applied with discrimina- 

 tion and knowledge of facts in order not to lead to a vanishing 

 summation, and a non sequitur conclusion as to the meaning 

 of the fluctuating departures and residuals in the observations. 

 It is proper that meteorologists should approach these summa- 

 tions cautiously. At present they are engaged in making their 

 records of observations homogeneous, and in studying the 

 general principles of inversion, radiation and absorption, under 

 the dominating influences of circulation. 



The Departures and Residuals of the Precipitation. 



Unfortunately it is more difficult to know how to discuss the 

 departures and residuals of the precipitation, because of the 

 greater irregularity in this element due to transportation of 



