320 A STUDY OF CORRELATIONS AMONG TERRESTRIAL TEMPERATURES. 



any length, but any tendency toward a period, may be shown. The period being 



regarded as entirely unknown, the observed departure from the general mean at any 



moment may be regarded as due to the periodic term which we seek, with accidental 



deviations superimposed. Let us put ao for the departure at some initial moment ; 



then let us take a series of equi-distant intervals of time, starting from the initial one, 



and let us put 



a„ a^, aj, • • ■, a,, 



the deviations at the ends of the intervals t, 2t, St, ■ ■ ■ nt. If there is any tendency 

 toward a rhythmical motion in these departures, having a period greater than 2t but 

 less than nt, then, in the general average, assuming ao to be positive, we should find 

 first a diminution and then an increase in the series of a's; that is, the curve repre- 

 senting tlie departures would be convex to the axis of abscissas. 



Since one departure may be taken for the initial one as well as another, we may 

 repeat this process with ai, aj, etc., as the initial departures, carrying the products in 

 each case to the requisite number of terms. We shall thus have a series of products 

 which may be continued as far as the series of observed departures extends. . Taking as 

 an example n — 5, the arrangement is the following : 



Sums : [aa] [aaj [aaj • • • [aaj 



This arrangement suggests the solution by least squares of a problem which may 

 be put into the following form. Starting as before with the initial departure ao, if the 

 fluctuation be a purely harmonic one, the departure at the end of lialf a period would 

 always be — ao, that at the end of a period +ao, etc. In general the departure at any 

 time t will be of the form ao + .r, x being a periodic function oft. Consequently the 

 actual deviations a^, aj, etc., will be of the form 



^1 = Vi =>= <'i ; ^2 = 3o*^2 ± ''2 ; etc. 



e,, C2, etc., being the purely accidental parts of the deviations. The problem thus 

 resolves itself into determining a series of factors x,, X2, X3, etc., by means of the 

 conditional equations 



a, = a^.T, ; a^ = a^.r^ ; etc. 



