| Angle of depression | 4.5° | 7.5° | 10.5° | 12.5° | 15.5°. |
| Relative frequency | 0.3 | 9.3 | 44.9 | 74.5 | 95.9. |
These figures are not wholly free from uncertainties, arising from true diurnal and annual variations in the frequency, but they give a good general idea of the influence of twilight.
If sunlight and twilight were the sole cause of the apparent annual variation, the frequency would have a simple period, with a maximum at midwinter and a minimum at midsummer. This is what is actually shown by the most northern stations and districts in Table I. When we come, however, below 65° lat. in Europe the frequency near the equinoxes rises above that at midwinter, and we have a distinct double period, with a principal minimum at midsummer and a secondary minimum at midwinter. In southern Europe—where, however, auroras are too few to give smooth results in a limited number of years—in southern Canada, and in the United States, the difference between the winter and summer months is much reduced. Whether there is any real difference between high and mean latitudes in the annual frequency of the causes rendered visible by aurora, it is difficult to say. The Scandinavian data, from the wealth of observations, are probably the most representative, and even in the most northern district of Scandinavia the smallness of the excess of the frequencies in December and January over those in March and October suggests that some influence tending to create maxima at the equinoxes has largely counterbalanced the influence of sunlight and twilight in reducing the frequency at these seasons.
5. Fourier Analysis.—With a view to more minute examination, the annual frequency can be expressed in Fourier series, whose terms represent waves, whose periods are 12, 6, 4, 3, &c. months. This has been done by Lovering (4) for thirty-five stations. The nature of the results will best be explained by reference to the formula given by Lovering as a mean from all the stations considered, viz.:—
8.33 + 3.03 sin(30t + 100°52′) + 2.53 sin(60t + 309° 5′) + 0.16 sin(90t + 213°31′) + 0.56 sin(120t + 162°45′) + 0.27 sin(150t + 32°38′).
The total number of auroras in the year is taken as 100, and t denotes the time, in months, that has elapsed since the middle of January. Putting t = 0, 1, &c., in succession, we get the percentages of the total number of auroras which occur in January, February, and so on. The first periodic term has a period of twelve, the second of six months, and similarly for the others. The first periodic term is largest when t × 30° + 100° 52′ = 450°. This makes t = 11.6 months after the middle of January, otherwise the 3rd of January, approximately. The 6-month term has the earliest of its two equal maxima about the 26th of March. These two are much the most important of the periodic terms. The angles 100° 52′, 309° 5′, &c., are known as the phase angles of the respective periodic terms, while 3.03, 2.53, &c., are the corresponding amplitudes. Table II. gives a selection of Lovering’s results. The stations are arranged according to latitude.
Plate I.
| Fig. 1—TWO TYPES OF AURORAL ARCS. |
| Fig. 2—TWO TYPES OF AURORAL RAYS. |
| (From the Internationale Polarforschung, 1882-1883, by permission of the Kaiserlichen Akademie der Wissenschaften, Vienna.) |
Plate II.
| Fig. 3—AURORAL BANDS. |
| Fig. 4—AURORAL CURTAIN BELOW AN ARC. |
| Fig. 5.—AURORAL CORONA. |