316 A STUDY OF CORRELATIONS AMONG TERRESTRIAL TEMPERATURES. 



temperature of each day, or through a period of several days, from the mean temper- 

 ature actually observed through the same period, we have a certain departure from 

 the normal, due to accidental or systematic causes. To fix the ideas I shall designate 

 the period for which the mean of these departures is taken as a time-term, or term 

 simply. The data then given by observation comprise the mean departures for a long 

 number of terms, each considered as a unit, and forming so far as possible a continuous 

 series. 



The most obvious classification of such departures is into periodic and irregular. 

 In the rigorous mathematical sense a periodic departure is one which always returns 

 to the same value at the end of an interval Pof time, called the period. This may be 

 either known or assumed in advance, or regarded as unknown. It cannot, however, 

 be determined as an unknown quantity from conditional equations, because it is im- 

 practicable so to introduce it as to give the equations a soluble form. If not regarded 

 as known we have to proceed by the method of trial and error. In this form the 

 question will be whether a certain assumed period P is indicated by observed depart- 

 ures. If the fluctuation had no other term than a purely periodic one as thus defined, 

 its existence could be ascertained by simple inspection. Imagining the fluctuations 

 to be expressed by the ordinates of a curve of which the abscissa is the time, we only 

 have to measure on the axis of abscissas from any arbitrary point, the series of distances 

 P, 2P, SP, etc., to the end of the series. We then take a number of intermediate 

 points and erect at each an ordinate expressing the observed departure. If P is the 

 true period the ordinates would have the same value at all the points distant from 

 each other by a multiple of P. Practically, however, we always have to deal with the 

 case in which other fluctuations than those of period P enter. We thus have acci- 

 dental deviations superposed upon the periodically recurring departures, which may 

 quite mask them. In this case it is necessary to take the mean value of the observed 

 departure at the several moments P, 2P, etc., after the initial moment. The mean of 

 all these values would be that corresponding to the initial phase. Taking, as an 

 example, the fluctuations represented in Figure (2), we see that the departure is 

 positive at the beginning of a period. 



The method of deciding whether a fluctuation of an assumed period P really 

 exists is this. We divide each period into any convenient number of equal parts 

 by the points 1, 2, 3, etc. We then taket he mean of all the ordinates at the several 

 points 1 ; the mean for the points 2, for the points 3, etc. The several means then 

 show the mean fluctuation during any one period. The absence of any fluctuation in 

 the given period would be shown l)y these mean values differing from each other 

 only by quantities which might be the result of the accidental deviations. 



