Number of occasions during 11 years when absolute range was:—  Means from the 5 largest
and 5 least ranges of the
month on the average of
11 years.
0′ to 5′. 5′ to 10′. 10′ to 15′. 15′ to 20′. 20′ to 25′. 25′ to 30′. 30′ to 35′. 35′ to 40′. over 40′. 5 largest. 5 least.
′ ′
January 51  145   69 37 24 7 4 3 1 22.90 5.07
February 26  99  84 51 26 10  4 2 8 27.21 6.55
March 1 72 138 61 32 21  8 1 7 29.87 8.93
April 0 43 167 73 27 10  6 3 1 23.69 10.31 
May 0 57 157 85 20 12  3 0 7 25.36 9.50
June 0 56 185 67 15 1 3 1 2 19.92 9.89
July 0 59 185 70 14 5 2 2 4 22.49 9.96
August 0 37 202 75 22 1 2 0 2 21.27 10.05 
September 1 68 153 71 19 5 4 5 4 24.55 9.52
October 3 103  111 67 34 10  11  2 0 23.92 8.01
November 42  140   81 28 14 9 8 5 3 23.58 5.64
December 64  166   56 29 14 7 1 1 3 20.43 4.36
  Totals 188  1045  1588  714  261  98  56  25  42 

§ 26. Magnetic phenomena, both regular and irregular, at any station vary from year to year. The extent of this variation is illustrated in Tables XXIV. and XXV., both relating to the period 1890 to 1900.[25] Table XXIV. gives the amplitudes of Relations to Sun-spot Frequency. the regular diurnal inequality in the elements stated at the head of the columns. The ordinary day declination data (D0) for Kew represent arithmetic means from the twelve months of the year; the other data all answer to the mean diurnal inequality for the whole year. Table XXV. gives the arithmetic means for each year of the absolute daily range, of the monthly range (or difference between the highest and lowest values in the month), and of the yearly range (or difference between the highest and lowest values of the year). The numerals attached to the years in these tables indicate their order as regards sun-spot frequency according to Wolf and Wolfer (see Aurora Polaris), 1893 being the year of largest frequency, and 1890 that of least. The difference in sun-spot frequency between 1897 and 1898 was microscopic; the differences between 1890, 1900 and 1899 were small, and those between 1893, 1894 and 1892 were not very large.

The years 1892-1895 represent high sun-spot frequency, while 1890, 1899 and 1900 represent low frequency. Table XXIV. shows that 1892 to 1895 were in all cases distinguished by the large size of the inequality ranges, and 1890, 1899 and 1900 by the small size. The range in 1893 is usually the largest, and though the H and V ranges at Ekaterinburg are larger in 1892 than in 1893, the excess is trifling. The phenomena apparent in Table XXIV. are fairly representative; other stations and other periods associate large inequality ranges with high sun-spot frequency. The diurnal inequality range it should be noticed is comparatively little influenced by irregular disturbances. Coming to Table XXV., we have ranges of a different character. The absolute range at Kew on quiet days is almost as little influenced by irregularities as is the range of the diurnal inequality, and in its case the phenomena are very similar to those observed in Table XXIV. As we pass from left to right in Table XXV., the influence of disturbance increases. Simultaneously with this, the parallelism with sun-spot frequency is less close. The entries relating to 1892 and 1894 become more and more prominent compared to those for 1893. The yearly range may depend on but a single magnetic storm, the largest disturbance of the year possibly far outstripping any other. But taking even the monthly ranges the values for 1893 are, speaking roughly, only half those for 1892 and 1894, and very similar to those of 1898, though the sun-spot frequency in the latter year was less than a third of that in 1893. Ekatarinburg data exactly analogous to those for Pavlovsk show a similar prominence in 1892 and 1894 as compared to 1893. The retirement of 1893 from first place, seen in the absolute ranges at Kew, Pavlovsk and Ekatarinburg, is not confined to the northern hemisphere. It is visible, for instance, in the amplitudes of the Batavia disturbance results. Thus though the variation from year to year in the amplitude of the absolute ranges is relatively not less but greater than that of the inequality ranges, and though the general tendency is for all ranges to be larger in years of many than in years of few sun-spots, still the parallelism between the changes in sun-spot frequency and in magnetic range is not so close for the absolute ranges and for disturbances as for the inequality ranges.

Table XXIV.—Ranges of Diurnal Inequalities.

Pavlovsk. Ekatarinburg. Kew.
D. I. H. D. I. H. V. Dq. Iq. Hq. Do.
′ ′ γ ′ ′ γ γ ′ ′ γ ′
189011 6.32 1.33 22 5.83 1.05 18  9 6.90 20 7.32
18916 7.31 1.79 30 6.85 1.38 25 14 8.04 1.52 28 8.48
18923 8.75 2.21 37 7.74 1.72 32 19 9.50 1.66 31 9.85
18931 9.64 2.24 38 8.83 1.80 31 17 10.06  1.96 35 10.74 
18942 8.58 2.17 38 7.80 1.73 30 17 9.32 1.94 34 9.80
18954 8.22 2.08 33 7.29 1.64 28 15 8.59 1.66 30 9.54
18965 7.39 1.77 29 6.50 1.38 25 15 7.77 1.31 25 8.50
18976 6.79 1.59 26 6.01 1.16 21 12 6.71 1.14 22 7.76
18987 6.25 1.56 26 5.76 1.19 21 11 6.85 1.07 21 7.59
18999 6.02 1.44 24 5.33 1.12 20 11 6.69 1.01 21 7.30
190010 6.20 1.28 22 5.88 0.93 17  8 6.52 1.06 21 6.83

Table XXV.—Absolute Ranges.

Kew Declination.
Daily.		 Pavlovsk.
Daily. Monthly. Yearly.
q. o. a. D. H. V. D. H. V. D. H. V.
′ ′ ′ ′ γ γ ′ γ γ ′ γ γ
189011 8.3 10.5 10.7 12.1 49 21 28.2 118  80 42.1 169 179
18916 10.0  12.8 13.7 16.0 70 39 46.3 218 233 92.3 550 614
18923 12.3  15.4 17.7 21.0 111  73 93.6 698 575 194.0  2416  1385 
18931 11.8  15.2 15.6 17.8 79 41 48.3 241 210 87.1 514 457
18942 11.3  14.7 16.5 20.4 97 62 84.1 493 493 145.6  1227  878
18954 10.6  14.8 15.6 18.1 80 46 47.4 220 223 73.9 395 534
18965 9.5 12.9 14.5 17.5 74 43 52.4 232 236 88.7 574 608
18978 8.2 11.5 12.1 14.6 61 30 43.8 201 170 101.1  449 480
18987 8.2 11.2 12.3 14.7 67 35 46.6 276 242 118.9  1136  888
18999 7.9 10.5 11.3 13.1 58 27 38.3 178 150 63.8 382 527
190010 7.4  8.9  9.2 10.5 44 16 32.8 134  89 94.2 457 365
Means 9.6 12.6 13.6 16.0 72 39 51.1 274 246 100.2  752 629

§ 27. The relationship between magnetic ranges and sun-spot frequency has been investigated in several ways. W. Ellis[26] has employed a graphical method which has advantages, especially for tracing the general features of the resemblance, and is besides independent of any theoretical hypothesis. Taking time for the axis of abscissae, Ellis drew two curves, one having for its ordinates the sun-spot frequency, the other the inequality range of declination or of horizontal force at Greenwich. The value assigned in the magnetic curve to the ordinate for any particular month represents a mean from 12 months of which it forms a central month, the object being to eliminate the regular annual variation in the diurnal inequality. The sun-spot data derived from Wolf and Wolfer were similarly treated. Ellis originally dealt with the period 1841 to 1877, but subsequently with the period 1878 to 1896, and his second paper gives curves representing the phenomena over the whole 56 years. This period covered five complete sun-spot periods, and the approximate synchronism of the maxima and minima, and the general parallelism of the magnetic and sun-spot changes is patent to the eye. Ellis[27] has also applied an analogous method to investigate the relationship between sun-spot frequency and the number of days of magnetic disturbance at Greenwich. A decline in the number of the larger magnetic storms near sun-spot minimum is recognizable, but the application of the method is less successful than in the case of the inequality range. Another method, initiated by Professor Wolf of Zurich, lends itself more readily to the investigation of numerical relationships. He started by supposing an exact proportionality between corresponding changes in sun-spot frequency and magnetic range. This is expressed mathematically by the formula

R = a + bS ≡ a {1 + (b/a) S },

where R denotes the magnetic range, S the corresponding sun-spot frequency, while a and b are constants. The constant a represents the range for zero sun-spot frequency, while b/a is the proportional increase in the range accompanying unit rise in sun-spot frequency. Assuming the formula to be true, one obtains from the observed values of R and S numerical values for a and b, and can thus investigate whether or not the sun-spot influence is the same for the different magnetic elements and for different places. Of course, the usefulness of Wolf’s formula depends largely on the accuracy with which it represents the facts. That it must be at least a rough approximation to the truth in the case of the diurnal inequality at Greenwich might be inferred from Ellis’s curves. Several possibilities should be noticed. The formula may apply with high accuracy, a and b having assigned values, for one or two sun-spot cycles, and yet not be applicable to more remote periods. There are only three or four stations which have continuous magnetic records extending even 50 years back, and, owing to temperature correction uncertainties, there is perhaps no single one of these whose earlier records of horizontal and vertical force are above criticism. Declination is less exposed to uncertainty, and there are results of eye observations of declination before the era of photographic curves. A change, however, of 1′ in declination has a significance which alters with the intensity of the horizontal force. During the period 1850-1900 horizontal force in England increased about 5%, so that the force requisite to produce a declination change of 19′ in 1900 would in 1850 have produced a deflection of 20′. It must also be remembered that secular changes of declination must alter the angle between the needle and any disturbing force acting in a fixed direction. Thus secular alteration in a and b is rather to be anticipated, especially in the case of the declination. Wolf’s formula has been applied by Rajna[28] to the yearly mean diurnal declination ranges at Milan based on readings taken twice daily from 1836 to 1894, treating the whole period together, and then the period 1871 to 1894 separately. During two sub-periods, 1837-1850 and 1854-1867, Rajna’s calculated values for the range differ very persistently in one direction from those observed; Wolf’s formula was applied by C. Chree[25] to these two periods separately. He also applied it to Greenwich inequality ranges for the years 1841 to 1896 as published by Ellis, treating the whole period and the last 32 years of it separately, and finally to all (a) and quiet (q) day Greenwich ranges from 1889 to 1896. The results of these applications of Wolf’s formula appear in Table XXVI.