A STUDY OF CORRELATIONS AMONG TERRESTRIAL TEMPERATURES. 317 



If the period is unknown, we must discover it tentatively by taking for P the 

 value which gives the best marked mean fluctuation, or the greatest range of value 

 among the mean departures. 



In the numerical computation on this principle, after the period is known, or has 

 been discovered, the most general mode of proceeding is that of development in a 

 Fourier series. We take an angle iV increasing uniformly with the time at such a rate 

 that it goes through 360° in the period P. Then, if we represent the departure at any 

 time by %\ we assume it, considered as a function of the time, to be developable in the 



form 



V = .v^ + x^ COS N + a-2 cos 2iV + ■ ■ ■ + y^sm N + y., sin 2N + ■ ■ • 



Regarding x^, a;,, Xj, • • • ^Vi> ^2. as unknown quantities, the coefficients of these 

 quantities at each epoch of observation will be the sines or the cosines in the second 

 number of the equation. Substituting for each moment of observation the values of 

 these sines and cosines, and taking the observed departures for v, we shall have a system 

 of equations for determining the unknowns. The solution of these equations by the 

 method of least squares will give the values of the unknowns which best represent the 

 observations. 



This method is sometimes employed in meteorology to determine and express 

 the diurnal and annual fluctuations in the temperature. For reasons not necesssary to 

 detail at present, the method of forming the mean values, in the manner first set forth, 

 and then finding the curve which best fits them, is preferable except when, for any 

 reason, all multiples of iV above the first are omitted. In this last case the fiuctuation 

 will be a purely harmonic one, the coefficients of which can be determined with great 

 facility by equations of condition. An example will be given in investigating the 

 fluctuations in temperature having the sun-spot period. 



§ 2. Irregular Fludications Tending Toiuard a Definite Period, — the Method of Time, Corre- 

 lation. 

 There is a class of fluctuations in which the period may be fairly definite, but yet 

 for which the preceding method would give no period whatever. This occurs when 

 we have a superposition of two classes of causes, or two sources of departure, one of 

 which, by itself, would result in a fluctuation in a definite period, while the other is 

 in the nature of perturbations, resulting in disturbances of the phase either continu- 

 ously or from time to time, and leading to seeming frequent changes in the length of 

 the period. If the preceding or any other method resting on the assumption of 

 unchanging period be applied to this case, the result might be that no period what- 

 ever would give a definite fluctuation. In other words a series of departures taken at 



A. p. S.— XXI. RR. 13, 1, '08. 



