curves, is characteristic of the winter months at Kew and Falmouth.
The shape is less variable in summer than in winter; but even in
summer the portion answering to the hours 6 p.m. to 6 a.m. varies
a good deal. The object of presenting the Kew and Falmouth
curves side by side is to emphasize the close resemblance between
the magnetic phenomena at places in similar latitudes, though
over 200 miles apart and exhibiting widely different ranges for
their meteorological elements. With considerable change of latitude
however the shape of vector diagrams changes largely.
Table XV.—Diurnal Inequalities in N. and W. at Falmouth (unit 1γ).
| Hour. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
| N. | a.m. | + 6 | + 5 | + 5 | + 5 | + 6 | + 6 | + 5 | + 1 | − 6 | −14 | −20 | −20 |
| p.m. | −17 | −12 | − 6 | − 1 | + 3 | + 6 | + 9 | + 9 | + 9 | + 8 | + 7 | + 7 | |
| W. | a.m. | − 2 | − 2 | − 3 | − 4 | − 6 | − 9 | −13 | −17 | −19 | −13 | − 3 | +11 |
| p.m. | +20 | +22 | +17 | +11 | + 6 | + 4 | + 2 | + 1 | 0 | − 1 | − 2 | − 2 | |
§ 20. Any diurnal inequality can be analysed into a series of harmonic terms whose periods are 24 hours and submultiples thereof. The series may be expressed in either of the equivalent Fourier Series. forms:—
| a1 cos t + b1 sin t + a2 cos 2t + b2 sin 2t + . . . | (i) |
| c1 sin (t + α1) + c2 sin (2t + α2) + . . . . | (ii) |
Table XVI.—Ranges in Diurnal Inequalities at Falmouth (unit 1γ).
| Jan. | Feb. | March. | April. | May. | June. | July. | Aug. | Sept. | Oct. | Nov. | Dec. | |
| N. | 21 | 23 | 30 | 39 | 39 | 37 | 37 | 39 | 36 | 32 | 24 | 15 |
| W. | 20 | 24 | 46 | 54 | 55 | 55 | 54 | 56 | 51 | 39 | 24 | 15 |
In both forms t denotes time, counted usually from midnight, one hour of time being interpreted as 15° of angle. Form (i) is that utilized in actually calculating the constants a, b, . . . Once the a, b, . . . constants are known, the c, α, . . . constants are at once derivable from the formulae:—
The a, b, c, α constants are called sometimes Fourier, sometimes Bessel coefficients.
 |
| (From Phil. Trans.) |
| Fig. 7. |
By taking a sufficient number of terms a series can always be obtained which will represent any set of diurnal inequality figures; but unless one can obtain a close approach to the observational figures from the terms possessing the periods 24, 12, 8 and 6 hours the physical significance and general utility of the analysis is somewhat problematical. In the case of the magnetic elements, the 24 and 12 hour terms are usually much the more important; the 24–hour term is generally, but by no means always, the larger of the two. The c constants give the amplitudes of the harmonic terms or waves, the α constants the phase angles. An advance of 1 hour in the time of occurrence of the first (and subsequent, if any) maximum and minimum answers to an increase of 15° in α1 of 30° in α2, of 45° in α3, of 60° in α4 and so on. In the case of magnetic elements the phase angles not infrequently possess a somewhat large annual variation. It is thus essential for a minute study of the phenomena at any station to carry out the analysis for the different seasons of the year, and preferably for the individual months. If the a and b constants are known for all the individual months of one year, or for all the Januarys of a series of years, we have only to take their arithmetic means to obtain the corresponding constants for the mean diurnal inequality of the year, or for the diurnal inequality of the average January of the series of years. This, however, is obviously not true of the c or α constants, unless the phase angle is absolutely unchanged throughout the contributory months or years. This is a point requiring careful attention, because when giving values of c and α for the whole year some authorities give the arithmetic mean of the c’s and α’s calculated from the diurnal inequalities of the individual months of the year, others give the values obtained for c and α from the mean diurnal inequality of the whole year. The former method inevitably supplies a larger value for c than the latter, supposing α to vary with the season. At some observatories, e.g. Greenwich and Batavia, it has long been customary to publish every year values of the Fourier coefficients for each month, and to include other elements besides the declination. For a thoroughly satisfactory comparison of different stations, it is necessary to have data from one and the same epoch; and preferably that epoch should include at least one 11–year period. There are, however, few stations which can supply the data required for such a comparison and we have to make the best of what is available. Information is naturally most copious for the declination. For this element E. Engelenburg[1] gives values of C1, C2, C3, C4, and of α1, α2, α3, α4 for each month of the year for about 50 stations, ranging from Fort Rae (62° 6′ N. lat.) to Cape Horn (55° 5′ S. lat.). From the results for individual stations, Engelenburg derives a series of means which he regards as representative of 11 different zones of latitude. His data for individual stations refer to different epochs, and some are based on only one year’s observations. The original observations also differ in reliability; thus the results are of somewhat unequal value. The mean results for Engelenburg’s zones must naturally have some of the sources of uncertainty reduced; but then the fundamental idea represented by the arrangement in zones is open to question. The majority of the data in Table XVII. are taken from Engelenburg, but the phase angles have been altered so as to apply to westerly declination. The stations are arranged in order of latitude from north to south; in a few instances results are given for quiet days. The figures represent in all cases arithmetic means derived from the 12 monthly values. In the table, so far as is known, the local mean time of the observatory has been employed. This is a point requiring attention, because most observatories employ Greenwich time, or time based on Greenwich or some other national observatory, and any departure from local time enters into the values of the constants. The data for Victoria Land refer to the “Discovery’s” 1902–1903 winter quarters, where the declination, taken westerly, was about 207°.5.
As an example of the significance of the phase angles in Table XVII., take the ordinary day data for Kew. The times of occurrence of the maxima are given by t + 234° = 450° for the 24-hour term, 2t + 39°.7 = 90° or = 450° for the 12-hour term, and so on, taking an hour in t as equivalent to 15°.
Thus the times of the maxima are:—
24-hour term, 2 h. 24 m. p.m.; 12-hour term, 1 h. 41 m. a.m. and p.m.
8-hour term, 4 h. 41 m. a.m., 0 h. 41 m. p.m., and 8 h. 41 m. p.m.
6-hour term, 0 h. 33 m. a.m. and p.m., and 6 h. 33 m. a.m. and p.m.
- ↑ Zur täglichen Variation der mag. Deklination (aus Heft II. des Archivs des Erdmagnetismus) (Potsdam, 1906).
