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effects of secular change and annual inequality. Eliminating the secular change, we arrive at an annual inequality, based on all days of the year including the highly disturbed. It is this annual inequality which appears under the heading s. It is certainly very unlike the annual inequality derived in the usual way. Whether the difference is to be wholly assigned to the fact that highly disturbed days contribute in the one case, but not in the other, is a question for future research.

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, also in the case of the horizontal f0rc%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. § 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 Abs I 1 not vary very much with the season, but still it does Ranoue vary. The principal minimum of the day, for instance, ge 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 TABLE XX.-Annual Inequality.

Declination. Inclination.

January -0-25 +0-04 +0-01 +0-08 -I-0-03 +0-32 -I-0-23 -l-0-06 -|-0-49 +O~32 +0-44 -0-03 February -0-54 -0 II 0-00 -I-0-48 +0-25 -o-20 -l-O~I9 -I-O'29 -i-0-39 +0-56 +0-29 -0-07 March -O'27 +0 04 +0-17 +0~03 -I-0-05 - 1-02 -0-12 -|-o-27 +0-20 +0-38 +0-13 +053 April -0-03 -I-0 IO +O°I2 -O~3I -0-14 -0-90 -0-11 +0-30 -o-08 -0-02 -O~I3 +0~18 - May +0-19 +007 -0-11 -0-39 -0-28 +0-29 -0-30 +0-08 -0'43 -0-29 -O-37 -0-15 June +0-46 +0 13 -0-14 -0-47 -0-39 +0-78 -O~I3 -0-19 -o-70 -0-77 -0-59 -0-35 July +0-48 +0 14 -0-17 -0-30 -°O'I3 -I-o-44 -0-08 -0-44 -0°72 -0-67 -0-27 -0-13 August +0-47 +0 II +0-01 +0-08 +0-05 -|-0-52 -O°I8 —0-38 -~0'47 -0-23 -0-05 -0-19 September +0-31 -l-O-OI 0-00 -i-0-29 -{-0-24 -0-02 +0-06 -0-06 -0-06 +0-16 +0-01 +O'2O October -0-07 -0 II +0-O9 -I-0-06 +0-01 -0-26 -|-0~03 -O°O4, -|-0-31 +0-27 +0-.19 0-00 November -0-30 -0 28 -0-05 +0-17 +0-1 1 -0-02 -1-0-08 -O°OI -|-o-51 +0-30 -I-0-43 +0-18 December -0-36 -0 14 +0-05 +0-26 +0-23 +0-05 -4-0-35 -{-0-06 +0-55 -}-0-19 +0-24 —0-29 p Range 1-02 042 0-34 0-95 0-64 1-So 0-65 0-74 1-27 1-33 1-03 0-88, 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 Ludeling 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.

§ 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 sub multiples. Fourier series can ":/"';“Z' also be formed representing the annual variation in the Fan; “Zo amplitudes of the regular diurnal inequality, and its 633515 component 24-hour, I2-hour, &c. waves, or of the amplitude of tha'abs0lute 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 I2 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-1-01)-I-P2 sin (2t-I-02)+ . . .,

where l is time counted from the beginning of the year, one month being taken as the equivalent of 30°, Pl, P2 represent the amplitudes, and 01, 62 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.” In the case of P1 and P2 the 'unit is I' for D and I, and 1-y for H and V. M denotes the mean value of the range for the 12 months. The letters g and 0 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. Increase in 0, or 02 means an earlier occurrence of the maximum or maxima, 1° answering roughly to one day in the case of the 12month term, and to half a day in the case of the 6-month term. PHM and P2/Nl 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 fro1n quiet days to ordinary days, or from years of sun spot minimum to years of sun spot maximum, which appears in the table, appears sallum. 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 dillferent 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 II 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 I2 months of the year (the eleven Januarys being combined into one, TABLE XXI.-Annual Variation of Diurnal Inequality Range. Fourier Coefficients.

Pl. P2. 01. 92. P1/M P2/M

Kew D0 3-36 0-94 279° 280° YO'4O 0-11

1890-1900 Dq 3-81 1 22 275° 273° 0-47 0-15

L, 0-67 O'I6 264° 269° O'42 0-10

H., 13-6 3-0 269° 261° 0-4 0-11

Vg 11-7 2 2 282° 242° 0-63 O'I2

Smax. Kew 4-50 1 25 277° K 282° I-70-473 0-13

D, Falmouth 4~I0 1 40 277° 286° 0'43 0'I5

Smin. 'Kew 3'35Y K1-'10 274° 269° -BT49 0-16

D, Falmouth 3-19 1-14 275° 277° 0-49 0-17

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 suliice 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