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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^[1] 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^[2] 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^[3] 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^[4] 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.

The Milan results are suggestive rather of heterogeneity in the material than of any decided secular change in a or b. The Greenwich data are suggestive of a gradual fall in a, and rise in b, at least in the case of the declination.

Table XXVII. gives values of a, b and b/a in Wolf’s formula calculated by Chree^[4] for a number of stations. There are two sets of data, the first set relating to the range from the mean diurnal inequality for the year, the second to the arithmetic mean of the ranges in the mean diurnal inequalities for the twelve months. It is specified whether the results were derived from all or from quiet days.

Table XXVI.—Values of a and b in Wolf’s Formula.

Milan.			Greenwich.
Epoch.	Declination  (unit 1′).		Epoch.	Declination  (unit 1′).		Horizontal Force  (unit 1γ).
	a.	b.		a.	b.	a.	b.
1836–94	5.31	.047	1841–96	7.29	.0377	26.4	.190
1871–94	5.39	.047	1865–96	7.07	.0396	23.6	.215
1837–50	6.43	.041	1889–96(a)	6.71	.0418	23.7	.218
1854–67	4.62	.047	1889–96(q)	6.36	.0415	25.0	.213

As explained above, a would represent the range in a year of no sun-spots, while 100 b would represent the excess over this shown by the range in a year when Wolf’s sun-spot frequency is 100. Thus

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1. P.T. 171. p. 541; P.R.S. 63, p. 64.
2. R.A.S. Notices 60, p. 142.
3. Rendiconti del R. Ist. Lomb. 1902, Series II. vol. 35.
4. P.T. 203 A, p. 151.