| c1. | c2. | c3. | c4. | α1. | α2. | α3. | α4. | ||
| ° | ° | ° | ° | ||||||
| I | Winter | 0.240 | 0.222 | 0.104 | 0.076 | 250.0 | 91.8 | 344 | 194 |
| Equinox | 0.601 | 0.290 | 0.213 | 0.127 | 290.3 | 135.5 | 4 | 207 | |
| Summer | 0.801 | 0.322 | 0.172 | 0.070 | 312.5 | 155.5 | 39 | 238 | |
| H | Winter | 3.62 | 3.86 | 1.81 | 1.13 | 82.9 | 277.3 | 154 | 6 |
| Equinox | 10.97 | 5.87 | 3.32 | 1.84 | 109.6 | 303.5 | 167 | 16 | |
| Summer | 14.85 | 6.23 | 2.35 | 0.95 | 130.3 | 316.5 | 199 | 41 | |
| V | Winter | 2.46 | 1.67 | 0.86 | 0.42 | 153.9 | 300.8 | 108 | 280 |
| Equinox | 6.15 | 4.70 | 2.51 | 0.94 | 117.2 | 272.3 | 99 | 289 | |
| Summer | 8.63 | 6.45 | 2.24 | 0.55 | 122.0 | 272.4 | 100 | 285 | |
In the case of the inclination, Liznar found that in both hemispheres the dip (north in the northern, south in the southern hemisphere) was larger than the normal when the sun was in perihelion, corresponding to an enhanced value of the horizontal force in summer in the northern hemisphere.
In the case of annual inequalities, at least that of the declination, it is a somewhat suggestive fact that the range seems to become less as we pass from older to more recent results, or from shorter to longer periods of years. Thus for Paris from 1821 to 1830 Arago deduced a range of 2′ 9″. Quiet days at Kew from 1890 to 1894 gave a range of 1′.2, while at Potsdam Lüdeling got a range 30% larger than that in Table XX. when considering the shorter period 1891-1899. Up to the present, few individual results, if any, can claim a very high degree of certainty. With improved instruments and methods it may be different in the future.
Table XX.—Annual Inequality.
| Declination. | Inclination. | |||||||||||
| Liznar, N. Hemi- sphere. | Potsdam, 1891-1906. | Parc St Maur, 1888-1897. | Kew (1890-1900). | Batavia, 1883-1893. | Mauritius. | Liznar & Hann’s mean. | Potsdam. | Parc St. Maur. | Kew. | |||
| q. | o. | s. | ||||||||||
| ′ | ′ | ′ | ′ | ′ | ′ | ′ | ′ | ′ | ′ | ′ | ′ | |
| January | −0.25 | +0.04 | +0.01 | +0.08 | +0.03 | +0.32 | +0.23 | +0.06 | +0.49 | +0.32 | +0.44 | −0.03 |
| February | −0.54 | −0.11 | 0.00 | +0.48 | +0.25 | −0.20 | +0.19 | +0.29 | +0.39 | +0.56 | +0.29 | −0.07 |
| March | −0.27 | +0.04 | +0.17 | +0.03 | +0.05 | −1.02 | −0.12 | +0.27 | +0.20 | +0.38 | +0.13 | +0.53 |
| April | −0.03 | +0.10 | +0.12 | −0.31 | −0.14 | −0.90 | −0.11 | +0.30 | −0.08 | −0.02 | −0.13 | +0.18 |
| May | +0.19 | +0.07 | −0.11 | −0.39 | −0.28 | +0.29 | −0.30 | +0.08 | −0.43 | −0.29 | −0.37 | −0.15 |
| June | +0.46 | +0.13 | −0.14 | −0.47 | −0.39 | +0.78 | −0.13 | −0.19 | −0.70 | −0.77 | −0.59 | −0.35 |
| July | +0.48 | +0.14 | −0.17 | −0.30 | −0.13 | +0.44 | −0.08 | −0.44 | −0.72 | −0.67 | −0.27 | −0.13 |
| August | +0.47 | +0.11 | +0.01 | +0.08 | +0.05 | +0.52 | −0.18 | −0.38 | −0.47 | −0.23 | −0.05 | −0.19 |
| September | +0.31 | +0.01 | 0.00 | +0.29 | +0.24 | −0.02 | +0.06 | −0.06 | −0.06 | +0.16 | +0.01 | +0.20 |
| October | −0.07 | −0.11 | +0.09 | +0.06 | +0.01 | −0.26 | +0.03 | −0.04 | +0.31 | +0.27 | +0.19 | 0.00 |
| November | −0.30 | −0.28 | −0.05 | +0.17 | +0.11 | −0.02 | +0.08 | −0.01 | +0.51 | +0.30 | +0.43 | +0.18 |
| December | −0.36 | −0.14 | +0.05 | +0.26 | +0.23 | +0.05 | +0.35 | +0.06 | +0.55 | +0.19 | +0.24 | −0.29 |
| Range | 1.02 | 0.42 | 0.34 | 0.95 | 0.64 | 1.80 | 0.65 | 0.74 | 1.27 | 1.33 | 1.03 | 0.88 |
§ 23. The inequalities in Table XX. may be analysed—as has in fact been done by Hann—in a series of Fourier terms, whose periods are the year and its submultiples. Fourier series can also be formed representing the annual variation in the Annual Variation Fourier Coefficients. amplitudes of the regular diurnal inequality, and its component 24-hour, 12-hour, &c. waves, or of the amplitude of the absolute daily range (§ 24). To secure the highest theoretical accuracy, it would be necessary in calculating the Fourier coefficients to allow for the fact that the “months” from which the observational data are derived are not of uniform length. The mid-times, however, of most months of the year are but slightly displaced from the position they would occupy if the 12 months were exactly equal, and these displacements are usually neglected. The loss of accuracy cannot be but trifling, and the simplification is considerable.
The Fourier series may be represented by
P1 sin (t + θ1) + P2 sin (2t + θ2) + ...,
where t is time counted from the beginning of the year, one month being taken as the equivalent of 30°, P1, P2 represent the amplitudes, and θ1, θ2 the phase angles of the first two terms, whose periods are respectively 12 and 6 months. Table XXI. gives the values of these coefficients in the case of the range of the regular diurnal inequality for certain specified elements and periods at Kew[23] and Falmouth.[23a] In the case of P1 and P2 the unit is 1′ for D and I, and 1γ for H and V. M denotes the mean value of the range for the 12 months. The letters q and o represent quiet and ordinary day results. S max. means the years 1892-1895, with a mean sun spot frequency of 75.0. S min. for Kew means the years 1890, 1899 and 1900 with a mean sun spot frequency of 9.6; for Falmouth it means the years 1899-1902 with a mean sun spot frequency of 7.25.