I 



A STUDY OF CORRELATIONS AMONG TERRESTRIAL TE>rPERATURES. 315 



we should find in the long run that there was no permanent correlation between the 

 fluctuations at the one place and at the other. For example, calHng the two regions 

 A and B, if we put into one class all the daj's on which the temperature in region A 

 is markedly above the normal, and in another class all the days in which it is mark- 

 edly below normal ; and if we take the temperatures in the distant region B for the 

 Same two classes of days, then, in the absence of any correlation, we should find the 

 mean temperatures at B to be the same in the two classes. If we found that the mean 

 temperature at B was above the normal when it was above the normal in A, and 

 below it in the contrary case, it would show that there was some common cause affect- 

 ing the two places. Should the mean temperature in B be entirely independent of 

 that in A it would show that there was no common cause affecting the temperature of 

 the two places and therefore that the fluctuations were not due to changes in the sun's 

 radiation. 



By this criterion the existence of either periodic. or non-periodic changes can be 

 equally well established, provided that a sufiiciently long series of observations is 

 made use of But it does not enable us to determine the laAV of change, but only the 

 general fact. When the general form of the law is known, especially when the fluc- 

 tuations are of definite period, other methods may be applied. 



CHAPTER I. 



IMethods of Investigating Fluctuating Quantities. 

 § 1. Fluctuations in a Fixed Period. 



The quantities with which we are concerned in the present paper are in the nature 

 of observed departures from normal or mean values. Such departures may be either 

 results of observation, or they may be derived a priori from some theory which is to 

 be tested by observation. Those considered in the present paper are of the first class. 

 We shall take up the general problem of studying fluctuations by considering it in the 

 form suggested by the special problem now before us. 



At every place and in every region on the surface of the earth there is for every 

 day a certain mean temperature, best determined by reading the thermometer at a 

 number of equi-distant intervals. These means may be extended through periods of 

 any length, thus giving a series of tempei-atures extending indefinitely year after year. 

 The temperatures thus observed undergo fluctuations in an annual period, which may 

 be represented either by a Fourier series, or by a smoothed curve extending as nearly 

 as may be through all the observed temperatures. A normal mean temperature for 

 each day throughout the year at any one place may thus be determined from the 

 observations of a number of years — the more the better. Subtracting the normal 



