Page:EB1911 - Volume 17.djvu/383

The minima, or extreme easterly positions in the waves, lie midway between successive maxima. All four terms, it will be seen, have maxima at some hour between 0h. 30m. and 2h. 30m. p.m. They thus reinforce one another strongly from 1 to 2 p.m., accounting for the prominence of the maximum in the early afternoon.

(From Phil. Trans.)

Fig. 8.

The utility of a Fourier analysis depends largely on whether the several terms have a definite physical significance. If the 24-hour and 12-hour terms, for instance, represent the action of forces whose distribution over the earth or whose seasonal variation is essentially different, then the analysis helps to distinguish these forces, and may assist in their being tracked to their ultimate source. Suppose, for example, one had reason to think the magnetic diurnal variation due to some meteorological phenomenon, e.g. heating of the earth’s atmosphere, then a comparison of Fourier coefficients, if such existed, for the two sets of phenomena would be a powerful method of investigation.

Table XVII.—Amplitudes and Phase Angles for Diurnal Inequality of Declination.

Place.	Epoch.	c1.	c2.	c3.	c4.	α1.	α2.	α3.	α4.
		′	′	′	′	°	°	°	°
Fort Rae (all)	1882–1883	18.49	8.22	1.99	2.07	156.5	41.9	308	104
Fort Rae (quiet)	”	9.09	4.51	1.32	0.73	166.5	37.5	225	350
Ekatarinburg	1841–1862	2.57	1.81	0.73	0.22	223.3	7.4	204	351
Potsdam	1890–1899	2.81	1.90	0.83	0.31	239.9	32.6	237	49
Kew (ordinary)	1890–1900	2.91	1.79	0.79	0.27	234.0	39.7	239	57
Kew (quiet)	”	2.37	1.82	0.90	0.30	227.3	42.1	240	55
Falmouth (quiet)	1891–1902	2.18	1.82	0.91	0.29	226.2	40.5	238	56
Parc St Maur	1883–1899	2.70	1.87	0.85	0.30	238.6	32.5	235	95
Toronto	1842–1848	2.65	2.34	1.00	0.33	213.7	34.9	238	350
Washington	1840–1842	2.38	1.86	0.65	0.33	223.0	26.6	223	53
Manila	1890–1900	0.53	0.58	0.43	0.17	266.3	50.7	226	89
Trivandrum	1853–1864	0.54	0.46	0.29	0.10	289.0	49.6		114
Batavia	1883–1899	0.80	0.88	0.43	0.13	332.0	163.2	5	236
St. Helena	1842–1847	0.68	0.61	0.63	0.34	275.8	171.4	27	244
Mauritius	1876–1890	0.86	1.11	0.76	0.22	21.6	172.7	350	161
C. of G. Hope	1841–1846	1.15	1.13	0.80	0.35	287.7	156.0	351	193
Melbourne	1858–1863	2.52	2.45	1.23	0.35	27.4	176.7	9	193
Hobart	1841–1848	2.29	2.15	0.87	0.32	33.6	170.8	349	185
S. Georgia	1882–1883	2.13	1.28	0.76	0.31	30.3	185.3	7	180
Victoria Land (all)	1902–1903	20.51	4.81	1.21	1.32	158.7	306.9	292	303
Victoria Land (quieter)	”	15.34	4.05	1.24	1.18	163.8	312.9	261

§ 21. Fourier coefficients of course often vary much with the season of the year. In the case of the declination this is especially true of the phase angles at tropical stations. To enter on details for a number of stations would unduly occupy space. A fair idea of the variability in the case of declination in temperate latitudes may be derived from Table XVIII., which gives monthly values for Kew derived from ordinary days of an 11-year period 1890–1900.

Fourier analysis has been applied to the diurnal inequalities of the other magnetic elements, but more sparingly. Such results are illustrated by Table XIX., which contains data derived from quiet days at Kew from 1890 to 1900. Winter includes November to February, Summer May to August, and Equinox the remaining four months. In this case the data are derived from mean diurnal inequalities for the season specified. In the case of the c or amplitude coefficients the unit is 1′ for I (inclination), and 1γ for H and V (horizontal and vertical force). At Kew the seasonal variation in the amplitude is fairly similar for all the elements. The 24-hour and 12–hour terms tend to be largest near midsummer, and least near midwinter; but the 8-hour and 6-hour terms have two well-marked maxima near the equinoxes, and a clearly marked minimum near midsummer, in addition to one near midwinter. On the other hand, the phase angle phenomena vary much for the different elements. The 24-hour term, for instance, has its maximum earlier in winter than in summer in the case of the declination and vertical force, but the exact reverse holds for the inclination and the horizontal force.

Table XVIII.—Kew Declination: Amplitudes and Phase Angles
(local mean time).

Month.	c1.	c2.	c3.	c4.	α1.	α2.	α3.	α4.
	′	′	′	′	°	°	°	°
January	1.79	0.86	0.41	0.27	251.2	29.8	254	64
February	2.41	1.11	0.57	0.30	242.0	27.7	235	39
March	3.05	1.98	1.11	0.45	233.2	36.1	223	49
April	3.35	2.48	1.17	0.39	224.8	39.2	228	61
May	3.57	2.38	0.87	0.17	221.3	50.8	245	89
June	3.83	2.39	0.74	0.05	212.6	46.7	239	72
July	3.72	2.30	0.77	0.11	214.6	48.1	233	8
August	3.64	2.43	1.05	0.18	228.2	57.2	244	51
September	3.35	2.02	1.04	0.35	236.9	55.3	245	70
October	2.69	1.69	0.92	0.48	240.1	35.6	235	65
November	1.94	1.06	0.51	0.32	248.3	28.3	247	61
December	1.61	0.81	0.35	0.20	255.1	22.0	243	56

§ 22. If secular change proceeded uniformly throughout the year, the value E_n of any element at the middle of the nth month of the year would be connected with E, the mean value for the whole year, by the formula E_n = E + (2n − 13)s/24, where s is the secular change per annum. For the present Annual Inequality.purpose, difference in the lengths of the months may be neglected. If one applies to E_n − E the correction −(2n − 13)s/24 one eliminates a regularly progressive secular change; what remains is known as the annual inequality. If only a short period of years is dealt with, irregularities in the secular change from year to year, or errors of observation, may obviously simulate the effect of a real annual inequality. Even when a long series of years is included, there is always a possibility of a spurious inequality arising from annual variation in the instruments, or from annual change in the conditions of observation. J. Liznar,^[1] from a study of data from a number of stations, arrived at certain mean results for the annual inequalities in declination and inclination in the northern and southern hemispheres, and J. Hann^[2] has more recently dealt with Liznar’s and newer results. Table XX. gives a variety of data, including the mean results given by Liznar and Hann. In the case of declination + denotes westerly position; in the case of inclination it denotes a larger dip (whether the inclination be north or south). According to Liznar declination in summer is to the west of the normal position in both hemispheres. The phenomena, however, at Parc St Maur are, it will be seen, the exact opposite of what Liznar regards as normal; and whilst the Potsdam results resemble his mean in type, the range of the inequality there, as at Parc St Maur, is relatively small. Of the three sets of data given for Kew the first two are derived in a similar way to those for other stations; the first set are based on quiet days only, the second on all but highly disturbed days. Both these sets of results are fairly similar in type to the Parc St Maur results, but give larger ranges; they are thus even more opposed to Liznar’s normal type. The last set of data for Kew is of a special kind. During the 11 years 1890 to 1900 the Kew declination magnetograph showed to within 1′ the exact secular change as derived from the absolute observations; also, if any annual variation existed in the position of the base lines of the curves it was exceedingly small. Thus the accumulation of the daily non-cyclic changes shown by the curves should closely represent the combined 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.

Table XIX.—Kew Diurnal Inequality: Amplitudes and Phase
Angles (local mean time).

		c1.	c2.	c3.	c4.	α1.	α2.	α3.	α4.
						°	°	°	°
I	Winter	0.240	0.222	0.104	0.076	250.0	91.8	344	194
	Equinox	0.601	0.290	0.213	0.127	290.3	135.5	4	207
	Summer	0.801	0.322	0.172	0.070	312.5	155.5	39	238
H	Winter	3.62	3.86	1.81	1.13	82.9	277.3	154	6
	Equinox	10.97	5.87	3.32	1.84	109.6	303.5	167	16
	Summer	14.85	6.23	2.35	0.95	130.3	316.5	199	41
V	Winter	2.46	1.67	0.86	0.42	153.9	300.8	108	280
	Equinox	6.15	4.70	2.51	0.94	117.2	272.3	99	289
	Summer	8.63	6.45	2.24	0.55	122.0	272.4	100	285

1. M.Z. 1888, 5, p. 225.
2. M.Z. 1904, 21, p. 129.