A STUDY OF CORRELATIONS AMONG TERRESTRIAL TEMPERATURES. 321 



These may be combined by the method of least squares. The normal equation is 



from which .r^ is at once found. Thus, putting i — 1, 2, 3, etc., we shall have a series 

 of quantities 



••■l' •'2> -'s' ' • .. 



of which numerical values may be determined from the equations. A tendency 

 toward a rhythmical deviation of the kind we are in search of will be shown by an 

 increasing value of x at the time corresponding most nearly to the completion of the 

 period. If there is no tendency toward any period between the limits 2t and nt the 

 series of xs will converge in the general mean toward the value zero. 



§ 3. Treatment of Fluctuations tvithout Discernible Period. 



The method developed in the tAvo preceding sections is applicable to a single series 

 of observations of fluctuating quantities of any kind, and will enable us to determine 

 any periodic tendency in them. We have now to consider the case in which the 

 periodicity is not discernible. In this case results are to be derived by comparisons of 

 different series of observations made simultaneously at different places. Our treat- 

 ment will be that of the special case of departures in temperature ; but the method 

 may of course be applicable in the wider field of fluctuating quantities in general. 



We know that deviations of the temperature from the normal are of constant 

 occurrence at every point of the globe. We also know that these are due, in great 

 part at least, to local causes, especially the motion of the air from region to region, and 

 the varying effects of cloud and moisture. But they may also be due in part to 

 changes in the sun's radiation of heat, or other general causes. The question is what 

 evidence can be found to indicate the action of a general cause affecting the whole 

 earth simultaneously. It is plain enough that observations at one place, no matter 

 hoAV long continued, would never enable us to distinguish bet^veen fluctuations of tem- 

 perature due to local causes and to the sun. But by comparing simultaneous observa- 

 tions in regions of the earth so widely separated that the same local causes could not 

 have influenced the temperatures in both regions, it is possible to determine, approxi- 

 mately at least, by a statistical method which we shall now develop, what part the sun 

 or other general cause may play in the fluctuations. 



The data for our problems are the simultaneous departures of temperature from 

 the normal, in a number n of regions, through a series of terms of equal length in 

 time, this length being chosen so as to best meet the requirements of the problem. 

 Let us put 



