318 A STUDY OF CORRELATIONS AMONG TERRESTRIAL TEMPERATURES. 



equi-distant moments would, in the long run, have for tlieir mean either an evanescent 

 value, or a constant value for all phases. 



To this class of fluctuations belong the ocean waves. If these are carefully 

 observed we shall generally And in them a tendency in a given state of the weather 

 to follow each other at fairly definite intervals, perhaps at 10, 15 or 20 seconds, accord- 

 ing to the distance between the crests. But should we take the mean period, how- 

 ever exact, and record the phase of the wave at any long series of moments separated 

 by exactly this period, we should find no one phase always recurring at the moments 

 thus defined. After a few seemingly regular recurrences of the wave, its height dimin- 

 ishes, perhaps almost to zero, or a fresh series of waves of similar period begins at a 

 different phase from that determined by the preceding waves. 



Another case of the same kind is afforded by the swing of a pendulum which is 

 subjected to a continually repeated disturbance, sometimes nearly stopping it, some- 

 times accelerating it, and sometimes changing the phase of the swing. How fre- 

 quently soever these disturbances may follow each other, there will always be in the 

 motion of the pendulum a tendency toward its regular period as a function of its 

 length. But it may be impracticable to determine any definite time of oscillation 

 through a long series of observations. In cases like these the perturbations may be so 

 considerable, and follow each other at such short intervals, and the regular mean 

 amplitude of the fluctuation may be so small or variable, that it will be impossible to 

 detect the tendency toward a regular period, except by the application of some special 

 method. To devise a method we must find some criterion for distinguishing between 

 a tendency toward a definite period and complete irregularity. 



Any tendency toward a definite period P may be defined in the following way : 

 Let T be the observed departure at any moment, and r' the departure at a definite 

 interval P following it. Now if there be really a tendency toward the period P, r' 

 should differ from t only by the difference of perturbations, or accidental deviations, 

 which may however be larger than either of the undisturbed departures, and there- 

 fore may completely mask the tendency toward equality between r and t'. However 

 this may be, the undisturbed departure midway of the period, that is, at the moment 

 *P, will have the opposite sign r and r'. It follows that in the general average, by 

 comparing a large number of departures in triplets, the individual members of which 

 are distant \P, and calling ti the mean of all the middle departui'es, the excess of 

 t' — T, will in the general average be opposite in sign from r' itself If a period be 

 found for which this holds true in the general mean, we have a tendency toward a 

 rhythmical movement in the period P. 



The detection of such a period is easy by a method which we may call that of 



