Increase in θ1 or θ2 means an earlier occurrence of the maximum or maxima, 1° answering roughly to one day in the case of the 12-month term, and to half a day in the case of the 6-month term. P1/M and P2/M both increase decidedly as we pass from years of many to years of few sun spots; i.e. relatively considered the range of the regular diurnal inequality is more variable throughout the year when sun spots are few than when they are many.

The tendency to an earlier occurrence of the maximum as we pass from quiet days to ordinary days, or from years of sun spot minimum to years of sun spot maximum, which appears in the table, appears also in the case of the horizontal force—at least in the case of the annual term—both at Kew and Falmouth. The phenomena at the two stations show a remarkably close parallelism. At both, and this is true also of the absolute ranges, the maximum of the annual term falls in all cases near midsummer, the minimum near midwinter. The maxima of the 6-month terms fall near the equinoxes.

Table XXI.—Annual Variation of Diurnal Inequality Range. Fourier Coefficients.

P1. P2. θ1. θ2. P1/M. P2/M.
Kew Do 3.36 0.94 279° 280° 0.40 0.11
1890-1900 Dq 3.81 1.22 275° 273° 0.47 0.15
Iq 0.67 0.16 264° 269° 0.42 0.10
Hq 13.6  3.0  269° 261° 0.48 0.11
Vq 11.7  2.2  282° 242° 0.63 0.12
S max. Kew 4.50 1.26 277° 282° 0.47 0.13
Dq Falmouth 4.10 1.40 277° 286° 0.43 0.15
S min. Kew 3.35 1.10 274° 269° 0.49 0.16
Dq Falmouth 3.19 1.14 275° 277° 0.49 0.17

§ 24. Allusion has already been made in § 14 to one point which requires fuller discussion. If we take a European station such as Kew, the general character of, say, the declination does not vary very much with the season, but still it does Absolute Range. vary. The principal minimum of the day, for instance, occurs from one to two hours earlier in summer than in winter. Let us suppose for a moment that all the days of a month are exactly alike, the difference in type between successive months coming in per saltum. Suppose further that having formed twelve diurnal inequalities from the days of the individual months of the year, we deduce a mean diurnal inequality for the whole year by combining these twelve inequalities and taking the mean. The hours of maximum and minimum being different for the twelve constituents, it is obvious that the resulting maximum will normally be less than the arithmetic mean of the twelve maxima, and the resulting minimum (arithmetically) less than the arithmetic mean of the twelve minima. The range—or algebraic excess of the maximum over the minimum—in the mean diurnal inequality for the year is thus normally less than the arithmetic mean of the twelve ranges from diurnal inequalities for the individual months. Further, as we shall see later, there are differences in type not merely between the different months of the year, but even between the same months in different years. Thus the range of the mean diurnal inequality for, say, January based on the combined observations of, say, eleven Januarys may be and generally will be slightly less than the arithmetic mean of the ranges obtained from the Januarys separately. At Kew, for instance, taking the ordinary days of the 11 years 1890-1900, the arithmetic mean of the diurnal inequality ranges of declination from the 132 months treated independently was 8′.52, the mean range from the 12 months of the year (the eleven Januarys being combined into one, and so on) was 8′.44, but the mean range from the whole 4,000 odd days superposed was only 8′.03. Another consideration is this: a diurnal inequality is usually based on hourly readings, and the range deduced is thus an under-estimate unless the absolute maximum and minimum both happen to come exactly at an hour. These considerations would alone suffice to show that the absolute range in individual days, i.e. the difference between the algebraically largest and least values of the element found any time during the 24 hours, must on the average exceed the range in the mean diurnal inequality for the year, however this latter is formed. Other causes, moreover, are at work tending in the same direction. Even in central Europe, the magnetic curves for individual days of an ordinary month often differ widely amongst themselves, and show maxima and minima at different times of the day. In high latitudes, the variation from day to day is sometimes so great that mere eye inspection of magnetograph curves may leave one with but little idea as to the probable shape of the resultant diurnal curve for the month. Table XXII. gives the arithmetic mean of the absolute daily ranges from a few stations. The values which it assigns to the year are the arithmetic means of the 12 monthly values. The Mauritius data are for different periods, viz. declination 1875, 1880 and 1883 to 1890, horizontal force 1883 to 1890, vertical force 1884 to 1890. The other data are all for the period 1890 to 1900.

Table XXII.—Mean Absolute Daily Ranges (Units 1′ for Declination, 1γ for H and V).

Jan.Feb.Mar.April.May.June.July.Aug.Sept.Oct.Nov.Dec.Year.
Declination.
Pavlovsk13.4217.2018.2217.2517.7615.9116.8916.5716.7515.7013.8712.37 15.99
Ekatarinburg 7.33 9.5411.9012.8913.6313.0312.7812.2111.23 9.44 7.866.8510.72
Kew. All days11.1613.6915.9315.0014.9013.6514.1314.2214.5714.0711.719.8013.57
Kew. Ordinary days10.1411.8714.1914.2413.8513.2613.4713.6713.7113.1010.409.0012.58
Kew. Quiet days 6.127.5710.5911.8412.0911.9511.6011.9310.86 9.166.545.08 9.61
Zi-ka-wei 3.88 3.25 6.22 7.04 7.15 7.40 7.77 8.06 6.73 4.68 2.912.52 5.63
Mauritius 6.93 7.79 7.11 5.75 4.87 4.03 4.36 6.00 6.28 6.71 6.996.78 6.13
Horizontal force.
Pavlovsk52.4 74.5 79.1 80.1 86.2 79.0 86.7 77.6 76.7 67.3 55.7 45.9 71.8 
Ekatarinburg33.2 43.1 48.4 51.7 56.2 54.1 56.7 51.7 49.3 44.1 34.1 29.3 46.0 
Mauritius37.9 35.0 36.2 37.6 35.0 34.1 33.8 34.5 36.6 37.4 37.8 35.3 35.9 
Vertical force.
Pavlovsk27.0 50.4 54.7 43.2 45.3 34.8 42.1 35.5 42.5 37.5 33.5 25.5 39.3 
Ekatarinburg17.4 26.6 29.2 30.1 29.6 27.6 29.6 26.1 25.2 22.1 19.6 16.4 24.9 
Mauritius17.1 19.5 20.1 17.3 16.5 15.5 17.1 22.0 22.7 19.4 16.7 15.2 18.2 

A comparison of the absolute ranges in Table XXII. with the inequality ranges for the same stations derivable from Tables VIII. to X. is most instructive. At Mauritius the ratio of the absolute to the inequality range is for D 1.38, for H 1.76, and for V 1.19. At Pavlovsk the corresponding ratios are much larger, viz. 2.16 for D, 2.43 for H, and 2.05 for V. The declination data for Kew in Table XXII. illustrate other points. The first set of data are derived from all days of the year. The second omit the highly disturbed days. The third answer to the 5 days a month selected as typically quiet. The yearly mean absolute range from ordinary days at Kew in Table XXII. is 1.49 times the mean inequality range in Table VIII.; comparing individual months the ratio of the absolute to the inequality range varies from 2.06 in January to 1.21 in June. Even confining ourselves to the quiet days at Kew, which are free from any but the most trifling disturbances, we find that the mean absolute range for the year is 1.20 times the arithmetic mean of the inequality ranges for the individual months of the year, and 1.22 times the range from the mean diurnal inequality for the year. In this case the ratio of the absolute to the inequality range varies from 1.55 in December to only 1.09 in May.

§ 25. The variability of the absolute daily range of declination is illustrated by Table XXIII., which contains data for Kew[24] derived from all days of the 11-year period 1890-1900. It gives the total number of times during the 11 years when the absolute range lay within the limits specified at the heads of the first nine columns of figures. The two remaining columns give the arithmetic means of the five largest and the five least absolute ranges encountered each month. The mean of the twelve monthly diurnal inequality ranges from ordinary days was only 8′.44, but the absolute range during the 11 years exceeded 20′ on 492 days, 15′ on 1196 days, and 10′ on 2784 days, i.e. on 69 days out of every 100.

Table XXIII.—Absolute Daily Range of Declination at Kew.