322 



A STUDY OF rORRELATIONS AMONO TERRESTRIAT. TEMPERATURKS. 



the n simultaneous departures of the temperature from the normal in the n regions for 

 any one term. 



Considering the problem thus suggested as that of determining the normal 

 departure of a world temperature, produced by any cause affecting the whole earth, 

 such as a change in the sun's radiation, the obvious method of determining this world 

 deviation is to take the mean of all the separate departures c,, observed in various 

 regions. Let us then put t, for the apparent mean departure of the world temperature 

 from the general normal. This appparent departure is determined by the equation 



or 



m = v^ + r^ + {>,+ •■• + r,^ = "E.v. 

 2.«. 



I t 



n 



0) 



(( = 1, 2, ■■•, h) 



Before taking up the question of a cosmical cause affecting the world-temperature, 

 let us consider the problem as that of determining the probable error of the departure 

 of the world-temperature from the normal. To do this we subtract t ifrom the indi- 

 vidual deviations, v^. We then have a series of residuals, ?(,, tt.,, etc. 



(( ^ «' — T 



Following the method of least squares let us form the squares of these residuals 



?t/ = r,- — 'Jt!', -I- T- 



Mj^ = r,^ — 2Ti'., -|- T" 



M ^ = t) " — 2tV -4- T" 

 n n n ' 



Putting e for the mean deviation and summing these residuals we shall have by 

 the theory of least squares the probable equation 



(n- l)e==S,u/ 



(2) 



Conceive now that we determine the deviation of the world-temperature in this 

 way through a number of time-terms, arriving at a series of values of t, each having 

 its mean error e^. It is clear that the value of the mean error should not be deter- 

 mined separately for each term from the discordances for that term alone, but from 

 the residuals throughout the whole period. If r be the entire number of time-terms 

 the number of these residuals will be nr. We represent by S, a summation through the 

 r terms, and by %j a summation of all the nr values. Then, by adding the r equations 

 of the form (2) we have the probable equation 



'•("-1)V = ^,/ 



(3) 



