with equal frequency it has been employed to suggest the recurrence of salient features of the various elements of weather and indeed to indicate the period of recurrence. We propose therefore to treat these types under separate headings as periodicity and curve-parallels. PERIODICITY A single curve representing the successive values of a meteorological element at successive times can be dealt with from the aspect of possible periodicity of occurrence of the values of the element; but the detection of periodicities is full of pitfalls for the unwary. The appearance of the curve of values itself is deceptive and the estimation of periods by cursory inspection, using the interval between prominences as a guide is apt to give quite illusory results. The difficulty arises from the interference of oscillations of different periods. For example if a quantity be subject to two oscillations concurrently, one of m years and the other of n years, the values will only recur in a number of years which is the least common multiple of both m and n, and if m and n have no common divisor, the period is mn years. In practice there is another difficulty arising from our ignorance of the life-history of the oscillation. It is usual tacitly to assume that a periodic oscillation in any meteorological element is due to some influence external to the oscillating system, which is of the same period as the oscillation itselfin other words that the oscillation is a forced oscillation1. But we are not entitled to assume that the oscillating system has no free period of its own. An oscillation in temperature, for example, may be regarded as an index of a periodic change in the general circulation of the atmosphere, and on the analogy of a typical vibrating system we must allow that if the general circulation is disturbed by some serious convulsion, an unusual extent of ice in the polar regions or a great volcanic eruption, the circulation will have natural periods of vibration of its own, and even if the disturbing cause is only temporary the oscillation, once set up, will take a long time to die away; and if the disturbing cause is itself periodic there will be all the complication that is incidental to the coexistence of forced and free vibrations. It seems impossible for us to determine the natural free periods of vibration of the circulation except by the long process of prolonged observations and the disentangling of the free and forced vibrations. To that, at present, we have hardly made an approach. In the present stage of the subject we have to determine the periods of the components of the vibrations as recorded in the observations, but while that stage is in progress we ought not rashly to assume that the phase of a vibration of particular period necessarily remains the same throughout the whole period of observation. In the analysis of the periodicity of sunspots, for example, the most conspicuous period for one group of observations is eleven years, yet if the analysis is extended over a hundred and fifty years the importance of the eleven year period becomes much reduced. That would certainly be the result if 1 The Theory of Sound, Rayleigh, Macmillan and Co., London, 1894, chaps. Iv and v. 2 'On Sunspot Periodicities,' Schuster, Proc. Roy. Soc., A, vol. LXXVII, 1906, pp. 141–45. the period were a natural period of the sun and was excited from time to time by a cause that was not strictly periodic in the same time. Fourier's theorem The fundamental theorem bearing upon the resolution of the natural variations of meteorological elements into periodic components is Fourier's theorem that any curve whatever representing variation with time can be resolved into a series of pure sine or cosine curves, that is to say of curves which are represented by the equation A1 its amplitude, a, the phase of its maximum, and so on for the other components. Harmonic analysis Fourier's theorem gives a theoretical method of determining A1, A2, A3, ... and α, α, α, .... How many terms will be necessary to represent any particular variation cannot be determined without trial. For meteorological purposes the analysis is sufficient if the combination of two, three or four terms gives a satisfying reproduction of the original curve. The process is applied generally to diurnal or seasonal variations and more particularly to mean values of those variations. In such cases the element may be supposed to be periodic within the day or the year, and the problem to be solved is to determine the constants A and a of the primary oscillation, and of its second, third, and fourth harmonic components. Here we may note that the form of component is not convenient for arithmetical computation; we can express it as This form is as general as the original one, but instead of a single component of specified amplitude and phase it represents the variation as consisting of two components of the same period one of which has its maximum when to and the other is zero at that epoch. If we assume that the variation which we are studying can be completely represented by a combination of cosine curves, a few observations will suffice for the determination of the constants. Take for example the seasonal variation of temperature at Jerusalem as indicated by alternate monthly values: Jan. 280, March 284.5, May 292.5, July 297, Sept. 295'5, Nov. 287. Assume that the variation is represented as Απί 4πt y = A。 + S1 sin + C1 cos + S2 sin + C2 cos 4πt; 2πt 2πt T T we desire to find S1, C1, S2, C2, t being measured from the middle of January. is the full period, namely one year. The interval of two months may be taken as one-sixth of the whole year, and as the sequence of events runs through 277 or 360° in the course of the year, the times of the successive values counting from the middle of January will be: We have to use the sines and cosines of these angles. For the sake of brevity let us call sin 60°, s, and cos 60°, c, instead of the numerical values √3/2 and 1/2. Then if the two cosine curves represent the variation adequately, remembering that the values of the sines and cosines will always be s or c, but with different signs for different quadrants, we get the following six equations for determining S1, C1, S2, C2: = s, and add. Thus we I'59. 8.42. We next multiply equation (0) by sin o°, (1) by sin 60°, s, (2) by sin 120°, s, (3) by sin 180°, o, (4) by sin 240°, – s, (5) by sin 300°, obtain 4s S1 = yn sin n × 60°, or S1 = yn sin n × 60° X Next multiply the equations in like manner by the cosines instead of the sines and add; we get C1 (2 + 4c2) = Σyn cos n × 60°, or C1 = yn cos n × 60° Thirdly multiply the equations in like manner by the series of sines of o°, 240°, 360°, 480°, 600° and add; we get 4s2S, Zyn sin n × 120°, or S2 Lastly multiply in like manner by the series of cosines of o°, 120°, 240°, 360°, 480°, 600° and add; we get C2 (2 + 4c2) Lyn cos n x 120°, or C2 We now remember that we can put the relation in the form = = = '92. = .144. 120°, T A2 = √C22 + S22, tan a2 = S2/C2. Substituting the values which we have obtained for S1, C1, S2, C2 we get = For the first component the maximum will be when 2771/T 191° or eleven days after the middle of July. For the second component the maximum will be when 4πt/T = 351°; t/r = 351°/720° or 178 days from the middle of January. We have used only six values for the determination of two components. The values for the alternate six months could be used in like manner to give independent values which ought to agree with those we have obtained. But with twelve values we can find constants for four components by a process which is similar in principle to that which we have used but is too voluminous for presentation here. The details are explained in the various books on the treatment of observations. The analysis of a curve for unknown periodicities. Method of residuation The usual method of harmonic analysis which has been sketched in what precedes is only applicable when we are justified in assuming a definite interval such as the day or the year as the period within which the values will recur. That is not the case when we are dealing for example with a long series of annual values as of temperature or rainfall. In that case we may obtain some simplification by the "method of residuation" which consists in annihilating variations of selected period while leaving unchanged, or only slightly modified, the other periodic changes1. 2π We may take for example a component variation a, sin t, where n is the period of the component in years and t is the time in years measured from a node of that particular component. If we add the ordinates with an interval of m years we get for the component of period n in the new curve The amplitude is zero if n = 2m, and thus in a curve of complex periodicity a component n is eliminated by adding the ordinates with an interval of n/2 years. All the other components have their amplitudes altered by the factor 2 cos πm/n and the phase of each is put forward by m/2 years. This process serves very well to eliminate disturbing components leaving the residual variation to be more easily identified provided that the interval m can be selected at discretion. That would obviously be the case with a continuous curve, but when only annual values are available and its operation is limited to periods of years its utility is similarly limited. In our own experience the most striking example of the resolution of an apparently irregular curve into harmonic components is that of the curve for the yield of wheat in Eastern England from 1885-1906. The original curves and the harmonic components of the long period term of eleven years are shown in the figure which illustrates very well the point from which we 1 'An investigation of the Seiches of Loch Earn by the Scottish Lake Survey. Part I. Limnographic Instruments and Methods of Observation,' by Professor G. Chrystal, Trans. Roy. Soc. Edin., vol. XLV, part II, no. 14, 1906, pp. 382-87. started, namely the unwisdom of relying upon the prominences of a curve of long series as a means of detecting the periods which make up the resultant variation. The final test of periodicity is the actual recurrence of successive values after a complete period, but with complex periodicities such as occur with nearly all natural phenomena that form of examination is not to be expected to be free from difficulty. It is remarkable that in the case referred 1896 1906 1886 1898 1899 1889 1890 1891 1903 1893 1905 1897 1887 1888 1900 1901 1902 1892 1904 1894 1895 Fig. 111. Six harmonic components of an eleven years' period and their resultant curve, marked by a thickened line, with figures for the yield of wheat in Eastern England 1885-1906. The upper line of dates refers to the upper row of dots and the lower line to the lower row. to in fig. 111 a test of that form held in an extraordinary manner, the values repeating themselves after eleven years with an astonishing degree of accuracy until 1906, considering the crudity and the vicissitudes of the data. Moreover the way in which deviations on either side of one point in the curves compensated each other was equally astonishing. These remarkable |