122 



DIURNAL INEQUALITIES. FOURIER COEFFICIENTS. 



arithmetic means in addition to the values of c and a derived from the mean diurnal inequality for the 

 year. A warning may, however, be not superfluous to the effect that in individual months " accidental " 

 irregularities are apt to influence largely the amplitude and phase angle of the Fourier waves of shorter 

 period. When one derives a diurnal inequality by combining together the months forming a season or the 

 year, these "accidental " effects tend to neutralise one another and disappear, but they do not do so in the 

 case of an arithmetic mean of amplitudes derived from individual months. The arithmetic mean of the c's 

 from the 12 months is necessarily larger than (or at least not less than) the c assigned in the tables to the 

 " year " (i.e. to the c derived from the mean diurnal inequality for the year). The difference between the 

 two c's is greater the more variable the phase angle throughout the year. The variation from month to 

 month in the phase angles in the tables is partly, no doubt, natural (i.e. representative of the average of 

 years), but it undoubtedly arises in part from " accidental " disturbances and from observational 

 uncertainties. The " accidental " phenomena are especially apt to influence the waves of shorter period, 

 and this is no doubt partly the reason why the values of c derived from the arithmetic mean of the 

 12 monthly values and from the mean diurnal inequality for the year are relatively so much closer for the 

 24-hour wave than for the others. In the case of the shorter-period waves an arithmetic mean could not 

 in all cases be assigned for the phase angle. Provided this angle varies slowly and regularly throughout 

 the year, if we get, say, 359 and 1 as its values in two consecutive months, we know that we must regard 

 these either as 359 and 361, or as - 1 and + 1, in forming a mean. But when, as in Table XXXIV, 

 the values in three successive months are 200, 40, and - 77, it is by no means clear how best to 

 interpret the figures. 



The phase angles were all calculated out to the nearest minute, though none of them can really claim 

 that degree of accuracy. The phase angles for the 24-hour wave are shown as calculated; for the 12-hour 

 wave decimals of a degree are retained ; for the 8-hour and 6-hour waves the results are recorded only to 

 the nearest degree. In all cases local time is used, Midnight answering to t = 0. Declination, it will be 

 remembered, is counted positive when the angle woN of the figure on p. 101 is above its mean value for the 

 day ; while inclination is regarded as increasing when the needle approaches the vertical. 



An increase in a phase angle means that the maxima and minima of the corresponding wave occur 

 earlier in the day. An advance of 1 hour in time requires an increase of 15 in a b of 30 in a 2 , of 45 

 in 0.3, and of 60 in a 4 . 



TABLE XXXIIL Declination (All Days). Amplitudes (Unit 1') and Phase Angles. 



