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TABLE XXVII.-Values of a and b in Wolf 's Formula. Declination I inclination Horizontal Force Vertical Force (unit 1'). (unit 1'). ' (unit 17). (unit 1-y). Diurnal Inequality for the Year. a. b. 100 b/a. a. b. 100 b/a. a. b. 100 1, /a a 7, 100 b/a. Pavlovsk, 1890-1900 ..... all 5-74 '04OO -7o 1-24 'O126 1-01 20-7 -211 1-02 8-1 -265 3-26 Pavlovsk, 1890-1900 quiet 6-17 -0424 -69 20'6 -195 - 0-95 5-9 -027 0-46 Ekatarinburg,1890-1900 all 5-29 -0342 -65 0-93 -0105 1-13 16-8 '182 1-09 8-6 'II7 1-37 Irkutsk, ,, , all 4-82 -0358 -74 0-97 -0087 o-90 18-2 -190 1-04 6-5 -071 1-09 Kew, , quiet 6°I0 'O433 -71 0-87 -0125 1-45 18-1 -194 1-07 14-3 -081 0-56 Falmouth, 1891-1902 quiet 5-90 'O45I -76 . 20-1 -233 I'I6 Kolaba, 1894-1901 quiet 2-37 -0066 -28 31-6 -281 0-89 19-4 -072 0-37 Batavia, 1887-1898 all 2'47 -0179 -72 3-60 -0218 0-61 38-7 '274 0-71 30-1 -156 0-52 Mauritius . all 4-06 -0164 ~40 15-0 -096 0-64 11-9 -069 0-58 Mean from individual months:-

Pavlovsk, 1890-1900 all 6-81 -0446 -66 1-44 -0151 1-05 22-8 -243 1-07 9-7 -287 2-97 quiet 6-52 -0442 -68 22-2 -208 0-94 7-0 -044 0-63 Ekatarinburg, 1890-1900 all 6-18 -0355 -58 1-12 -0120 I-06 19-2 -195 1-01 9-2 -156 I-70 Greenwich, 1865-1896 all 7-07 -0396 -56 23-6 -215 0-91 Kew, 1890-1900 all 6-65 '0428 -64 . . . . quiet 6-49 -0410 -63 1-17 'O130 1-11 21-5 -191 0'89 I6°O -072 0-45 Falmouth, 1891-1902 quiet 6-16 '045O -73 20-9 -236 1-13 nv -i ' ' Jw:Z l I i l l l l l,

b/a seems the most natural measure of sun-spot influence. Accepting it, we see that sun-spot influence appears larger at most places for inclination and horizontal force than for declination. In the case of vertical force there is at Pavlovsk, and probably in a less measure at other northern stations, a large difference between all and quiet days, which is not shown in the other elements. The difference between the values of b/a at different stations is also exceptionally large for vertical force. Whether this last result is wholly free from observational uncertainties is, however, open to some doubt, as the agreement between /V0lf's formula and observation is in general somewhat inferior for vertical force. In the case of the declination, the mean numerical difference between the observed values and those derived from Wolf 's formula, employing the values of a and b given in Table XXVII., represented on the average about 4% of the mean value of the element for the period considered, the probable error representing about 6% of the difference between the highest and lowest values observed. The agreement was nearly, if not quite, as good as this for inclination and horizontal force, but for vertical force the corresponding percentages were nearly twice as large.

Applying Wolf's formula to the diurnal ranges for different months of the year, Chree found, as was to be anticipated, that the constant a had an annual period, with a conspicuous minimum at midwinter; but whilst b also varied, it did so to a much less extent, the consequence being that b/a showed a minimum at midsummer. The annual variation in b/a alters with the place, with the element, and with the type of day from which the magnetic data are derived. Thus, in the case of Pavlovsk declination, whilst the mean value of 100 b/a for the 12 months is, as shown in Table XXVII., 0-66 for all and 0-68 for quiet days-values practically identical-if we take the four midwinter and the four midsummer months separately, we have 100 b/a, varying from 0~8I in winter to 0-52 in summer on all days, but from I-39 in winter to 0'52 in summer on quiet days. In the case of horizontal force at Pavlovsk the corresponding figures to these are for all days-winter I'77, summer 0-98, but for quiet days -winter 1-83, summer 0-71.

W0lf's formula has also been applied to the absolute daily ranges, to monthly ranges, and to various measures of disturbance. In these cases the values found for b/a are usually larger than those found for diurnal ine uality ranges, but the accordance between observed values and fliose calculated from W0lf's formula is less good. If instead of the range of the diurnal inequality we take the sum of the 24-hourly differences from the mean for the day-or, what comes to the same thing, the average departure throughout the 24 hours from the mean value for the day-we find that the resulting Wolf's formula gives at least as good an agreement with observation as in the case of the inequality range itself. The formulae obtained in the case of the 24 differences, at places as wide apart as Kew and Batavia, agreed in giving a decidedly larger value for b/a than that obtained from the ranges. This indicates that the inequality curve is relatively less peaked in years of many than in years of few suns ots.

p§ 28. The applications of Ellis's and Wolf 's methods relate directly only to the amplitude of the diurnal changes. There is, however, a change not merely in amplitude but in type. This is clearly seen when we compare the values found in years of many and of few sunspots for the Fourier coefficients in the diurnal inequality. Such a comparison is carried out in Table XXVIII. for the declination on ordinary days at Kew. Local mean time is used. The heading S max. (sun-sp0t maximum) denotes mean average results from the four years 1 892-I 895, having a mean sun-spot frequency of 75-0, whilst S min. (sun-spot minimum) applies similarly to the years 1890, 1899 and 1900, having a mean sun-spot frequency of only 9-6. The data relate to the mean diurnal inequality for the whole year or for the season stated. It will be seen that the difference between the c, or amplitude, coefficients in the S max. and S min. years is greater for the 24-hour term than for the 12-hour term, greater for the I2hour than for the 8-hour term, and hardly apparent in the 6-hour term. Also, relatively considered, the difference between the amplitudes in S max. and S min. years is greatest in winter and least in summer. Except in the case of the 6-hour term, where the differences are uncertain, the phase angle is larger, Le. maxima and minima occur earlier in the day, in years of S min. than in years of S max. Taking the results for the whole year in Table XXVIII., this advance of phase in the S min. years represents in time I5-6 minutes for the 24-hour term, 9-4 minutes for the 12-hour term, and 14-7 minutes for the 8-hour term. The difference in the phase angles, as in the amplitudes, is greatest in winter. Similar phenomena are shown by the horizontal force, and at Falmouth 24 as well as Kew. TABLE XXVIII.-Fourier Coefficients in Years of many and few Sun-spots.

Year. Wint er. Equinox. I Summer.,

Smax. S min. S max. S II1iI'l.;S max. Smin S max. S min. I I I I I 7 I I I

ti 3'47 2-21 2-41 I'43 3-76 2-41 4-38,2'98

Q 2-04 1-51 1-15 0-78 2-33 1-71 2-73 2'06

ca 0-89 0-72 0-55 0-42 1-16 0-97 0-97 0-77

ai 0-28 0-27 0-30 0-27 0-42 0-42 0-11 0°II

0 To O 0 O 0 O O

al 228-5 232-4 243-0 256-0 231-3 233-7 218-2 220-3 G2 4¥°7 45'5 23'5 35'9 40'6 43'9 50'6 52'5

as 232-6 243-6 234-0 257-6 228-4 236-2 236-8 245-4 al 58-0 57~3 52-3 60-8 62-0 58-2 57-4 45-2

§ 29. There have already been references to quiet days, for instance in the tables of diurnal inequalities. It seems to have been originally supposed that quiet days differed from other days only I t D in the absence of irregular disturbances, and that mean Q" e ay annual values, or secular change data, or diurnal inequal- Phenamum ities, derived from them might be regarded as truly normal or representative of the station. It was found, however, by P. A. Müller W that mean annual values of the magnetic elements at St Petersburg and Pavlovsk from 1873 to 1885 derived from quiet days alone differed in a systematic fashion from those derived from all days, and analogous results were obtained by Ellis 3° at Greenwich for the period 1889*I 896. The average excesses for the quiet-day over the all-day means in these two cases were as follows:- Westerly - Horizontal Vertical*

Declination! InC1mat'0n'{ Force. l Force. I

St Petersburg +0-24 -0-23 +3-2-y -0-Sy

Greenwich +0-08 +3-3y -0-9-y


The sign of the difference in the case of D, I and H wasithe same in each year examined by Müller, and the same was true of H at Greenwich. In the case of V, and of D at Greenwich, the differences are