+--------+-------------+-------------+--_Encyclopaedia Britannica, 11th Edition, "McKinley, William" to "Magnetism, Terrestrial" Volume 17, Slice 3

+--------+-------------+-------------+-------+-------+--------+-------+-------+--------+-------+---------+

+--------+-------------+-------------+-------+-------+--------+-------+-------+--------+-------+---------+

| Period.| 1882-1883. | 1873-1885. |1890 |1891 |1883 |1888 |1894 |1883 |1884 | 1902 |

| | | | -1900.| -1900.| -1897. | -1898.| -1901.| -1894. | -1890.| -1903. |

+--------+------+------+------+------+-------+-------+--------+-------+-------+--------+-------+---------+

| | a. | q. | a. | q. | a. | q. | a. | a. | q. | a. | a. | a. |

+--------+------+------+------+------+-------+-------+--------+-------+-------+--------+-------+---------+

| Hour | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ |

| 1 | +4.6 | +1.5 | -0.5 | -0.3 | -0.4 | -0.3 | -0.3 | -0.1 | +0.6 | +0.9 | +0.3 | +0.6 |

| 2 | +5.0 | +1.6 | -0.5 | -0.3 | -0.3 | -0.2 | -0.3 | -0.1 | +0.6 | +0.8 | +0.2 | +0.7 |

| 3 | +5.6 | +1.6 | -0.5 | -0.3 | -0.3 | -0.2 | -0.3 | -0.1 | +0.5 | +0.6 | 0.0 | +0.7 |

| 4 | +5.0 | +1.5 | -0.4 | -0.3 | -0.3 | -0.2 | -0.4 | -0.2 | +0.5 | +0.5 | -0.0 | +0.7 |

| 5 | +4.2 | +1.4 | -0.5 | -0.3 | -0.2 | -0.2 | -0.4 | -0.2 | +0.7 | +0.3 | -0.1 | +0.7 |

| 6 | +2.4 | +1.2 | -0.4 | -0.3 | -0.1 | -0.1 | -0.3 | -0.1 | +0.8 | +0.1 | -0.2 | +0.5 |

| 7 | +0.7 | +0.9 | -0.2 | -0.1 | +0.2 | +0.1 | 0.0 | 0.0 | +0.5 | -0.2 | -0.3 | +0.4 |

| 8 | -0.1 | +0.8 | +0.1 | +0.3 | +0.6 | +0.4 | +0.4 | +0.3 | -0.2 | -0.8 | -0.4 | +0.3 |

| 9 | -0.7 | +0.8 | +0.6 | +0.6 | +1.0 | +0.8 | +0.7 | +0.5 | -1.2 | -1.7 | -0.4 | +0.1 |

| 10 | -1.2 | +0.9 | +1.0 | +1.0 | +1.1 | +1.0 | +0.9 | +0.3 | -1.9 | -2.7 | -0.5 | -0.2 |

| 11 | -2.2 | +0.8 | +1.2 | +1.2 | +1.0 | +0.9 | +0.7 | 0.0 | -2.1 | -3.3 | -0.6 | -0.4 |

| Noon | -3.4 | +0.4 | +1.1 | +1.1 | +0.6 | +0.6 | +0.4 | -0.5 | -1.6 | -3.1 | -0.7 | -0.7 |

| 1 | -4.5 | -0.2 | +0.7 | +0.7 | +0.3 | +0.2 | +0.2 | -0.6 | -0.8 | -2.4 | -0.8 | -0.9 |

| 2 | -5.6 | -1.2 | +0.4 | +0.4 | +0.1 | +0.1 | +0.2 | -0.5 | -0.2 | -1.3 | -0.6 | -1.0 |

| 3 | -6.3 | -2.2 | +0.2 | +0.1 | 0.0 | 0.0 | +0.2 | -0.3 | +0.3 | -0.2 | -0.3 | -1.0 |

| 4 | -6.1 | -2.9 | 0.0 | -0.1 | -0.1 | -0.1 | +0.2 | +0.1 | +0.3 | +0.7 | +0.1 | -0.9 |

| 5 | -5.1 | -3.2 | -0.1 | -0.3 | -0.2 | -0.2 | +0.1 | +0.4 | +0.2 | +1.3 | +0.4 | -0.7 |

| 6 | -3.1 | -2.9 | -0.2 | -0.3 | -0.3 | -0.3 | 0.0 | +0.5 | +0.2 | +1.5 | +0.5 | -0.5 |

| 7 | -1.7 | -2.2 | -0.3 | -0.4 | -0.4 | -0.4 | -0.2 | +0.4 | +0.3 | +1.6 | +0.5 | -0.2 |

| 8 | +0.3 | -1.3 | -0.3 | -0.5 | -0.4 | -0.4 | -0.3 | +0.2 | +0.4 | +1.6 | +0.6 | 0.0 |

| 9 | +2.0 | -0.3 | -0.4 | -0.6 | -0.4 | -0.4 | -0.3 | +0.1 | +0.5 | +1.6 | +0.6 | +0.2 |

| 10 | +2.5 | +0.5 | -0.5 | -0.6 | -0.4 | -0.3 | -0.3 | 0.0 | +0.6 | +1.5 | +0.6 | +0.4 |

| 11 | +3.0 | +1.0 | -0.5 | -0.6 | -0.4 | -0.3 | -0.3 | 0.0 | +0.6 | +1.4 | +0.5 | +0.5 |

| 12 | +4.0 | +1.3 | -0.5 | -0.4 | -0.4 | -0.3 | -0.3 | -0.1 | +0.6 | +1.2 | +0.4 | +0.6 |

+--------+------+------+------+------+-------+-------+--------+-------+-------+--------+-------+---------+

| Range | 11.9 | 4.8 | 1.7 | 1.8 | 1.5 | 1.4 | 1.3 | 1.1 | 2.9 | 4.9 | 1.4 | 1.7 |

+--------+------+------+------+------+-------+-------+--------+-------+-------+--------+-------+---------+

TABLE XII.--Diurnal Inequality of Declination (+ to West).

+--------+-----------+-----------+-----------+-----------+-----------+-----------+-----------+-----------+

+--------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+

| Month. |June.|Dec. |June.|Dec. |June.|Dec. |June.|Dec. |June.|Dec. |June.|Dec. |June.|Dec. |June.|Dec. |

+--------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+

| Hour | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ |

| 1 |-0.4 |-0.1 |-0.3 | 0.0 |-0.3 |-0.1 |+0.1 |+0.1 |-0.1 |-0.4 | 0.0 |+0.1 |-0.4 |-0.7 |+0.8 |+1.1 |

| 2 |-0.2 |+0.4 |-0.3 |+0.1 |-0.4 |+0.1 |-0.1 |+0.1 |-0.2 |-0.1 |-0.2 |+0.2 |-0.5 |-0.4 |+0.3 |+1.1 |

| 3 |-0.2 |-0.1 |-0.3 |+0.1 |-0.4 |+0.3 |-0.2 |+0.2 |-0.2 |+0.1 |-0.2 |+0.4 |-0.7 |-0.1 |-0.1 |+1.0 |

| 4 |-1.2 |-0.4 |-0.3 |+0.3 |-0.5 |+0.5 |-0.3 |+0.3 |-0.3 |+0.3 |-0.2 |+0.7 |-0.6 |+0.3 |-0.1 |+1.1 |

| 5 |-2.9 |-0.6 |-0.7 |+0.4 |-0.7 |+0.7 |-0.3 |+0.5 |-0.5 |+0.6 |-0.3 |+1.0 |-0.7 |+1.0 | 0.0 |+1.7 |

| 6 |-5.2 |-0.6 |-1.6 |+0.5 |-1.6 |+1.1 |-0.5 |+1.2 |-1.0 |+0.9 |-0.4 |+1.7 |-1.0 |+2.2 | 0.0 |+2.7 |

| 7 |-6.2 |-0.9 |-2.2 |+0.7 |-1.7 |+1.4 |-1.1 |+2.0 |-2.2 |+1.9 |-1.1 |+2.6 |-1.6 |+3.3 |-0.1 |+4.4 |

| 8 |-6.0 |-1.2 |-2.1 |+0.2 |-1.1 |+0.9 |-0.4 |+2.3 |-1.5 |+2.2 |-1.0 |+2.4 |-0.8 |+3.6 |+0.1 |+5.6 |

| 9 |-4.4 |-1.8 |-1.1 |-0.1 |-0.2 |+0.5 |+0.5 |+2.0 |-0.3 |+1.3 |+0.2 |+2.0 |+0.7 |+3.1 |+0.6 |+5.6 |

| 10 |-1.5 |-1.1 | 0.0 |-0.2 |+0.6 |+0.3 |+0.9 |+1.3 |+0.3 |+0.2 |+1.2 |+1.1 |+1.6 |+1.6 |+1.2 |+3.6 |

| 11 |+2.1 |+0.6 |+1.2 | 0.0 |+1.2 |+0.1 |+1.0 |+0.4 |+0.5 |-1.0 |+1.4 | 0.0 |+1.5 |+0.1 |+1.0 |+0.7 |

| Noon |+4.8 |+2.2 |+2.1 | 0.0 |+1.4 |-0.4 |+0.7 |-0.6 |+0.3 |-1.4 |+1.0 |-1.4 |+0.8 |-1.0 |-0.1 |-2.6 |

| 1 |+6.1 |+3.2 |+2.0 |-0.2 |+1.1 |-0.8 |+0.3 |-1.4 |+0.3 |-1.2 |+0.1 |-2.2 |+0.3 |-1.8 |-1.4 |-5.1 |

| 2 |+6.1 |+3.2 |+1.6 |-0.3 |+0.7 |-0.9 |-0.2 |-1.8 |+0.2 |-0.4 |-0.9 |-2.5 |-0.3 |-1.9 |-2.2 |-6.2 |

| 3 |+5.2 |+2.4 |+0.9 |-0.3 |+0.3 |-0.9 |-0.7 |-1.9 |+0.2 |+0.4 |-1.5 |-2.2 |-0.3 |-1.4 |-2.4 |-5.8 |

| 4 |+3.6 |+1.5 |+0.2 |-0.3 |+0.1 |-0.8 |-0.8 |-1.6 |+0.7 |+0.6 |-1.3 |-1.6 |+0.2 |-0.8 |-1.6 |-4.8 |

| 5 |+1.8 |+0.5 | 0.0 |-0.2 | 0.0 |-0.4 |-0.5 |-1.2 |+1.1 |+0.4 |-0.3 |-1.0 |+0.5 |-0.8 |-0.7 |-3.3 |

| 6 |+0.7 |-0.1 |+0.1 |-0.2 |+0.2 |-0.4 |-0.1 |-0.7 |+1.0 |+0.1 |+0.5 |-0.5 |+0.5 |-0.6 |-0.4 |-1.9 |

| 7 | 0.0 |-0.8 |+0.3 |-0.2 |+0.5 |-0.4 |+0.1 |-0.6 |+0.6 |-0.4 |+0.7 |-0.3 |+0.4 |-0.8 | 0.0 |-1.0 |

| 8 | 0.0 |-1.2 |+0.4 |-0.1 |+0.5 |-0.3 |+0.2 |-0.5 |+0.5 |-0.7 |+0.7 |-0.3 |+0.3 |-0.9 |+0.5 |-0.3 |

| 9 |-0.5 |-1.4 |+0.3 |-0.1 |+0.4 |-0.2 |+0.4 |-0.3 |+0.4 |-0.9 |+0.6 |-0.2 |+0.2 |-0.9 |+1.1 | 0.0 |

| 10 |-0.5 |-1.7 |+0.1 | 0.0 |+0.2 |-0.1 |+0.4 |-0.1 |+0.2 |-1.0 |+0.4 |-0.1 |+0.1 |-1.0 |+1.3 |+0.6 |

| 11 |-0.7 |-1.1 |-0.1 |-0.1 | 0.0 |-0.1 |+0.3 | 0.0 |+0.1 |-0.8 |+0.3 | 0.0 | 0.0 |-1.0 |+1.3 |+0.9 |

| 12 |-0.6 |-0.7 |-0.2 |-0.1 |-0.2 |-0.1 |+0.2 |+0.1 |-0.1 |-0.6 |+0.1 |+0.1 |-0.2 |-1.0 |+1.1 |+1.2 |

+--------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+

| Range |12.3 | 5.0 | 4.3 | 1.0 | 3.1 | 2.3 | 2.1 | 4.2 | 3.3 | 3.6 | 2.9 | 5.1 | 3.2 | 5.5 | 3.7 |11.8 |

+--------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+

§ 15. In the case of the horizontal force there are, as Table IX.

shows, two markedly different types of diurnal inequality. In the one

type, exemplified by Pavlovsk or Greenwich, the force is below its

mean value in the middle of the day; it has a principal minimum about

10 or 11 a.m., and morning and evening maxima, the latter usually the

largest. In the other type, exemplified by Kolaba or Batavia, the

horizontal force is above its mean in the middle of the day, and has

a maximum about 11 a.m. The second type may be regarded as the

tropical type. At tropical stations, such as Kolaba, Batavia, Manila

and St Helena, the type is practically the same in summer as in

winter, and is the same whether the station is north or south of the

equator. Similarly, what we may call the temperate type is seen--with

comparatively slight modifications--both in summer and winter at

stations such as Greenwich or Pavlovsk. In winter, it is true, the

pronounced daily minimum is a little later and the early morning

maximum is relatively more important than in summer. There is not, as

in the case of the declination, any essential difference between the

phenomena at temperate stations in the northern and southern

hemispheres.

[Illustration: FIG. 6.]

With diminishing latitude, there is a gradual transition from the

temperate to the tropical type of horizontal force diurnal variation,

and at stations whose latitude is under 45° there is a very

appreciable variation in type with the season. The mean diurnal

variation for the year at Tiflis in Table IX. really represents a

struggle between the two types, in which on the whole the temperate

type prevails. If we take the diurnal variations at Tiflis for

midsummer and midwinter, we find the former essentially of the

temperate, the latter essentially of the tropical type. A similar

conflict may be seen in the mean diurnal inequality for the year at

the Cape of Good Hope, but there the tropical type on the whole

predominates, and it prevails more at midwinter than at midsummer.

Toronto and Hobart, though similar in latitude to Tiflis, show a

closer approach to the temperate type. Still at both stations the

hours during which the force is below its mean value tend to extend

back towards midnight, especially at midsummer. The amplitude of the

horizontal force range appears less at intermediate stations, such as

Tiflis, than at stations in either higher or lower latitudes. There

is a very great difference in this respect between the north and the

south of India.

§ 16. In the case of the vertical force in higher temperate

latitudes--at Pavlovsk for instance--the diurnal inequalities from

"all" and from "quiet" days differ somewhat widely in amplitude and

slightly even in type. In mean latitudes, e.g. at Tiflis, there is

often a well marked double period in the mean diurnal inequality for

the whole year; but even at Tiflis this is hardly, if at all, apparent

in the winter months. In the summer months the double period is

distinctly seen at Kew and Greenwich, though the evening maximum is

always pre-eminent. Speaking generally, the time of the minimum, or

principal minimum, varies much less with the season than that of the

maximum. At Kew, for instance, on quiet days the minimum falls between

11 a.m. and noon in almost all the months of the year, but the time of

the maximum varies from about 4 p.m. in December to 7 p.m. in June. At

Kolaba the time of the minimum is nearly independent of the season;

but the changes from positive to negative in the forenoon and from

negative to positive in the afternoon are some hours later in winter

than in summer. At Batavia the diurnal inequality varies very little

in type with the season, and there is little evidence of more than one

maximum and minimum in the day. At Batavia, as at Kolaba, negative

values occur near noon; but it must be remembered that while at Kolaba

and more northern stations vertical force urges the north pole of a

magnet downwards, the reverse is true of Batavia, as the dip is

southerly. At St Helena vertical force is below its mean value in the

forenoon, but the change from - to + occurs at noon, or but little

later, both in winter and summer. At the Cape of Good Hope the

phenomena at midsummer are similar to those at Kolaba, the force being

below its mean value from about 9 a.m. to 3 p.m. and above it

throughout the rest of the day; but at midwinter there is a

conspicuous double period, the force being below its mean from 1 a.m.

to 7 a.m. as well as from 11 a.m. to 3 p.m., and thus resembling the

all-day annual results at Greenwich. At Hobart vertical force is below

its mean value from 1 a.m. to 9 a.m. at midsummer, and from 4 a.m. to

noon at midwinter; while the force is above its mean persistently

throughout the afternoon both in summer and winter, there is at

midwinter a well marked secondary minimum about 6 p.m., almost the

same hour as that at which the maximum for the day is observed in

summer.

TABLE XIII.--Range of the Diurnal Inequality of Declination.

+-------------+-------------+------+------+------+------+------+------+------+------+------+------+------+------+

| Place. | Period. | Jan. | Feb. |March.|April.| May. | June.| July.| Aug. | Sept.| Oct. | Nov. | Dec. |

+-------------+-------------+------+------+------+------+------+------+------+------+------+------+------+------+

| | | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ |

| Pavlovsk | 1890-1900 a | 4.93 | 6.15 | 8.58 |10.93 |12.18 |12.27 |11.82 |11.38 | 8.70 | 6.87 | 5.54 | 4.63 |

| " | " q | 2.96 | 4.20 | 8.73 |11.28 |12.89 |13.28 |12.31 |11.70 | 9.37 | 6.91 | 3.95 | 2.66 |

| Ekatarinburg| 1890-1900 a | 3.33 | 4.32 | 7.63 |11.19 |11.82 |11.58 |11.09 |10.45 | 8.13 | 5.60 | 3.73 | 3.14 |

| Greenwich | 1865-1896 a | 5.87 | 7.07 | 9.40 |11.42 |10.55 |10.90 |10.82 |10.93 | 9.66 | 8.15 | 6.41 | 5.15 |

| Kew | 1890-1900 a | 4.92 | 6.06 | 9.08 |10.95 |10.66 |10.92 |10.59 |11.01 | 9.49 | 7.73 | 5.37 | 4.46 |

| " | " q | 4.07 | 4.76 | 8.82 |10.57 |10.92 |10.62 |10.18 |11.01 | 9.76 | 7.51 | 4.75 | 3.34 |

| Toronto | 1842-1848 a | 5.96 | 6.05 | 9.18 | 9.94 |11.55 |12.34 |12.21 |13.14 |10.76 | 6.96 | 6.32 | 4.97 |

| Manila | 1890-1900 a | 1.79 | 1.09 | 2.13 | 3.02 | 3.84 | 3.94 | 4.21 | 4.89 | 4.53 | 1.83 | 0.85 | 1.33 |

| Trivandrum | 1853-1864 a | 2.06 | 1.48 | 0.79 | 1.67 | 2.90 | 3.06 | 3.06 | 3.64 | 3.31 | 1.27 | 2.14 | 2.33 |

| Batavia | 1884-1899 a | 4.18 | 4.64 | 3.57 | 2.93 | 2.38 | 2.03 | 2.31 | 3.16 | 3.80 | 4.51 | 4.50 | 4.19 |

| St Helena | 1842-1847 a | 3.72 | 5.19 | 4.93 | 3.30 | 2.64 | 3.24 | 3.42 | 3.59 | 2.40 | 4.43 | 4.05 | 3.54 |

| Mauritius | 1876-1890 a | 5.2 | 6.1 | 6.3 | 4.7 | 4.1 | 2.9 | 3.4 | 4.9 | 5.0 | 5.5 | 5.6 | 5.1 |

| Cape | 1841-1846 a | 5.14 | 8.21 | 7.27 | 5.00 | 3.91 | 3.21 | 3.54 | 4.98 | 4.33 | 5.96 | 6.36 | 5.47 |

| Hobart | 1841-1848 a |11.66 |11.80 | 9.50 | 7.26 | 4.56 | 3.70 | 4.61 | 5.89 | 8.24 |11.01 |12.05 |11.81 |

+-------------+-------------+------+------+------+------+------+------+------+------+------+------+------+------+

§ 17. Variations of inclination are connected with those of horizontal

and vertical force by the relation

[delta]I = ½ sin 2I {V^-1 [delta]V - H^-1 [delta]H}.

Thus in temperate latitudes where V is considerably in excess of H,

whilst diurnal changes in V are usually less than those in H, it is

the latter which chiefly dominate the diurnal changes in inclination.

When the H influence prevails, I has its highest values at hours when

H is least. This explains why the dip is above its mean value near

midday at stations in Table XI. from Pavlovsk to Parc St Maur. Near

the magnetic equator the vertical force has the greater influence.

This alone would tend to make a minimum dip in the late forenoon, and

this minimum is accentuated owing to the altered type of the

horizontal force diurnal variation, whose maximum now coincides

closely with the minimum in the vertical force. This accounts for the

prominence of the minimum in the diurnal variation of the inclination

at Kolaba and Batavia, and the large amplitude of the range. Tiflis

shows an intermediate type of diurnal variation; there is a minimum

near noon, as in tropical stations, but inclination is also below its

mean for some hours near midnight. The type really varies at Tiflis

according to the season of the year. In June--as in the mean equality

from the whole year--there is a well marked double period; there is a

principal minimum at 2 p.m. and a secondary one about 4 a.m.; a

principal maximum about 9 a.m. and a secondary one about 6 p.m. In

December, however, only a single period is recognizable, with a

minimum about 8 a.m. and a maximum about 7 p.m. The type of diurnal

inequality seen at the Cape of Good Hope does not differ much from

that seen at Batavia. Only a single period is clearly shown. The

maximum occurs about 8 or 9 p.m. throughout the year. The time of the

minimum is more variable; at midsummer it occurs about 11 a.m., but at

midwinter three or four hours later. At Hobart the type varies

considerably with the season. In June (midwinter) a double period is

visible. The principal minimum occurs about 8 a.m., as at the Cape.

But, corresponding to the evening maximum seen at the Cape, there is

now only a secondary maximum, the principal maximum occurring about 1

p.m. At midsummer the principal maximum is found--as at Kew or

Greenwich--about 10 or 11 a.m., the principal minimum about 4 p.m.

§ 18. Even at tropical stations a considerable seasonal change is

usually seen in the amplitude of the diurnal inequality in at least

one of the magnetic elements. At stations in Europe, and generally in

temperate latitudes, the amplitude varies notably in all the elements.

Table XIII. gives particulars of the inequality range of declination

derived from hourly readings at selected stations, arranged in order

of latitude from north to south. The letters "a" and "q" are used in

the same sense as before. At temperate stations in either

hemisphere--e.g. Pavlovsk, Greenwich or Hobart--the range is

conspicuously larger in summer than in winter. In northern temperate

stations a decided minimum is usually apparent in December. There is,

on the other hand, comparatively little variation in the range from

April to August. Sometimes, as at Kew and Greenwich, there is at least

a suggestion of a secondary minimum at midsummer. Manila and

Trivandrum show a transition from the December minimum, characteristic

of the northern stations, to the June minimum characteristic of the

southern, there being two conspicuous minima in February or March and

in November or October. At St Helena there are two similar minima in

May and September, while a third apparently exists in December. It

will be noticed that at both Pavlovsk and Kew the annual variation in

the range is specially prominent in the quiet day results.

Table XIV. gives a smaller number of data analogous to those of Table

XIII., comprising inequality ranges for horizontal force, vertical

force and inclination. In some cases the number of years from which

the data were derived seems hardly sufficient to give a smooth annual

variation. It should also be noticed that unless the same group of

years is employed the data from two stations are not strictly

comparable. The difference between the all and quiet day vertical

force data at Pavlovsk is remarkably pronounced. The general tendency

in all the elements is to show a reduced range at midwinter; but in

some cases there is also a distinct reduction in the range at

midsummer. This double annual period is particularly well marked at

Batavia.

TABLE XIV.--Ranges in the Diurnal Inequalities.

+-------------------------------+------+------+------+------+------+------+------+------+------+------+------+------+

| | Jan. | Feb. |March.|April.| May. | June.| July.| Aug. | Sept.| Oct. | Nov. | Dec. |

+-------------------------------+------+------+------+------+------+------+------+------+------+------+------+------+

| Pavlovsk 1890-1900 a | 12 | 20 | 32 | 46 | 47 | 49 | 49 | 44 | 39 | 32 | 17 | 11 |

| " " q | 12 | 17 | 31 | 42 | 45 | 45 | 42 | 40 | 37 | 31 | 17 | 10 |

| Ekatarinburg " a | 11 | 15 | 29 | 37 | 40 | 40 | 39 | 36 | 33 | 27 | 13 | 9 |

| Kew " q | 15 | 17 | 26 | 36 | 38 | 39 | 38 | 38 | 35 | 27 | 20 | 11 |

| Toronto 1843-1848 a | 23 | 21 | 24 | 28 | 29 | 29 | 26 | 28 | 41 | 25 | 21 | 20 |

| Batavia 1883-1898 a | 49 | 47 | 54 | 60 | 51 | 48 | 50 | 53 | 58 | 52 | 43 | 40 |

| St Helena 1843-1847 a | 43 | 41 | 48 | 53 | 46 | 40 | 40 | 45 | 41 | 40 | 40 | 32 |

| Mauritius 1883-1890 a | 21 | 15 | 21 | 23 | 20 | 21 | 20 | 22 | 20 | 21 | 21 | 20 |

| Cape of Good Hope 1841-1846 a | 13 | 10 | 13 | 13 | 15 | 16 | 14 | 18 | 21 | 14 | 17 | 20 |

| Hobart 1842-1848 a | 42 | 43 | 34 | 28 | 19 | 17 | 22 | 23 | 23 | 35 | 39 | 42 |

| | | | | | | | | | | | | |

| Pavlovsk 1890-1900 a | 15 | 27 | 29 | 24 | 26 | 20 | 23 | 19 | 23 | 20 | 18 | 14 |

| " " q | 4 | 5 | 9 | 13 | 13 | 12 | 13 | 10 | 9 | 7 | 5 | 4 |

| Ekatarinburg " a | 10 | 15 | 17 | 21 | 22 | 19 | 20 | 16 | 14 | 13 | 11 | 9 |

| Kew 1891-1900 q | 7 | 10 | 20 | 25 | 31 | 27 | 28 | 23 | 20 | 15 | 9 | 6 |

| Toronto 1843-1848 a | 12 | 14 | 17 | 23 | 26 | 14 | 27 | 32 | 34 | 25 | 19 | 18 |

| Batavia 1883-1898 a | 42 | 48 | 48 | 45 | 31 | 31 | 32 | 29 | 41 | 50 | 40 | 33 |

| St Helena 1843-1847 a | 16 | 13 | 12 | 14 | 13 | 11 | 17 | 11 | 17 | 11 | 15 | 18 |

| Mauritius 1884-1890 a | 12 | 16 | 18 | 15 | 14 | 13 | 15 | 21 | 20 | 16 | 13 | 11 |

| Cape of Good Hope 1841-1846 a | 29 | 47 | 41 | 38 | 21 | 12 | 14 | 19 | 19 | 35 | 33 | 28 |

| Hobart 1842-1848 a | 25 | 27 | 22 | 23 | 24 | 21 | 22 | 28 | 26 | 22 | 23 | 27 |

| | | | | | | | | | | | | |

| _Inclination_ | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ |

| Pavlovsk 1890-1900 a | 0.97 | 1.24 | 2.07 | 2.79 | 2.72 | 2.88 | 2.85 | 2.64 | 2.52 | 2.18 | 1.20 | 0.89 |

| Ekatarinburg " a | 0.79 | 0.94 | 1.70 | 2.08 | 2.25 | 2.19 | 2.18 | 2.08 | 2.00 | 1.70 | 0.88 | 0.69 |

| Kew " q | 0.98 | 1.01 | 1.38 | 1.86 | 2.05 | 2.02 | 2.05 | 2.15 | 1.98 | 1.57 | 1.27 | 0.63 |

| Toronto 1843-1848 a | 1.15 | 0.94 | 1.19 | 1.23 | 1.31 | 1.37 | 1.13 | 1.26 | 1.87 | 1.16 | 1.09 | 1.05 |

| Batavia 1883-1898 a | 4.88 | 5.22 | 5.56 | 5.62 | 4.21 | 4.05 | 4.24 | 4.17 | 5.13 | 5.58 | 4.51 | 3.85 |

| Cape of Good Hope 1842-1846 a | 1.55 | 2.29 | 2.23 | 2.23 | 1.60 | 1.41 | 1.54 | 1.70 | 1.86 | 2.03 | 1.55 | 2.04 |

| Hobart 1842-1848 a | 1.95 | 2.16 | 1.72 | 1.62 | 1.23 | 1.16 | 1.28 | 1.42 | 1.39 | 1.75 | 2.04 | 2.10 |

+-------------------------------+------+------+------+------+------+------+------+------+------+------+------+------+

§ 19. When discussing diurnal inequalities it is sometimes convenient

to consider the components of the horizontal force in and

perpendicular to the astronomical meridian, rather than the horizontal

force and declination. If N and W be the components of H to

astronomical north and west, and D the westerly declination, N = H

cos D, W = H sin D. Thus corresponding small variations in N, W, H and

D are connected by the relations:--

[delta]N = cos D[delta]H - H sin D[delta]D,

[delta]W = sin D[delta]H + H cos D[delta]D.

If [delta]H and [delta]D denote the departures of H and D at any hour

of the day from their mean values, then [delta]N and [delta]W

represent the corresponding departures of N and W from their mean

values. In this way diurnal inequalities may be calculated for N and W

when those for H and D are known. The formulae suppose [delta]D to be

expressed in absolute measure, i.e. 1´ of arc has to be replaced by

0.0002909. If we take as an example a station at which H is .185 then

H[delta]D = .0000538 (number of minutes in [delta]D). In other words,

employing 1[gamma] as unit of force, one replaces H[delta]D by

5.38[delta]D, where [delta]D represents declination change expressed

as usual in minutes of arc. In calculating diurnal inequalities for N

and W, one ought, strictly speaking, to assign to H and D the exact

mean values belonging to these elements for the month or the year

being dealt with. For practical purposes, however, a slight departure

from the true mean values is immaterial, and one can make use of a

constant value for several successive years without sensible error. As

an example, Table XV. gives the mean diurnal inequality for the whole

year in N and W at Falmouth, as calculated from the 12 years 1891 to

1902. The unit employed is 1[gamma].

The data in Table XV. are closely similar to corresponding Kew data,

and are presumably fairly applicable to the whole south of England for

the epoch considered. At Falmouth there is comparatively little

seasonal variation in the type of the diurnal variation in either N or

W. The amplitude of the diurnal range varies, however, largely with

the season, as will appear from Table XVI., which is based on the same

12 years as Table XV.

Diurnal inequalities in N and W lend themselves readily to the

construction of what are known as _vector diagrams_. These are curves

showing the direction and intensity at each hour of the day of the

horizontal component of the disturbing force to which the diurnal

inequality may be regarded as due. Figs. 7 and 8, taken from the

_Phil. Trans._ vol. 204A, will serve as examples. They refer to the

mean diurnal inequalities for the months stated at Kew (1890 to 1900)

and Falmouth (1891 to 1902), thick lines relating to Kew, thin to

Falmouth. NS and EW represent the geographical north-south and

east-west directions; their intersection answers to the origin (thick

lines for Kew, thin for Falmouth). The line from the origin to M

represents the magnetic meridian. The line from the origin to any

cross--the number indicating the corresponding hour counted from

midnight as 0--represents the magnitude and direction at that hour of

the horizontal component of the disturbing force to which the diurnal

inequality may be assigned. The cross marks the point whose

rectangular co-ordinates are the values of [delta]N and [delta]W

derived from the diurnal inequalities of these elements. In figs. 7

and 8 the distances of the points N, E, S, W from their corresponding

origin represents 10[gamma]. The tendency to form a loop near

midnight, seen in the November and December curves, is

characteristic of the winter months at Kew and Falmouth. The shape is

less variable in summer than in winter; but even in summer the portion

answering to the hours 6 p.m. to 6 a.m. varies a good deal. The object

of presenting the Kew and Falmouth curves side by side is to emphasize

the close resemblance between the magnetic phenomena at places in

similar latitudes, though over 200 miles apart and exhibiting widely

different ranges for their meteorological elements. With considerable

change of latitude however the shape of vector diagrams changes

largely.

TABLE XV.--Diurnal Inequalities in N. and W. at Falmouth (unit

1[gamma]).

+---------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+

| Hour. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

+---------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+

|N. /a.m. | + 6 | + 5 | + 5 | + 5 | + 6 | + 6 | + 5 | + 1 | - 6 | -14 | -20 | -20 |

| \p.m. | -17 | -12 | - 6 | - 1 | + 3 | + 6 | + 9 | + 9 | + 9 | + 8 | + 7 | + 7 |

|W. /a.m. | - 2 | - 2 | - 3 | - 4 | - 6 | - 9 | -13 | -17 | -19 | -13 | - 3 | +11 |

| \p.m. | +20 | +22 | +17 | +11 | + 6 | + 4 | + 2 | + 1 | 0 | - 1 | - 2 | - 2 |

+---------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+

Fourier Series.

§ 20. Any diurnal inequality can be analysed into a series of harmonic

terms whose periods are 24 hours and submultiples thereof. The series

may be expressed in either of the equivalent forms:--

a1 cos t + b1 sin t + a2 cos 2t + b2 sin 2t + ... (i)

c1 sin (t + [alpha]1) + c2 sin (2t + [alpha]2) + .... (ii)

TABLE XVI.--Ranges in Diurnal Inequalities at Falmouth (unit

1[gamma]).

+----+------+------+------+------+------+------+------+------+------+------+------+------+

| | Jan. | Feb. |March.|April.| May. | June.| July.| Aug. | Sept.| Oct. | Nov. | Dec. |

+----+------+------+------+------+------+------+------+------+------+------+------+------+

| N. | 21 | 23 | 30 | 39 | 39 | 37 | 37 | 39 | 36 | 32 | 24 | 15 |

| W. | 20 | 24 | 46 | 54 | 55 | 55 | 54 | 56 | 51 | 39 | 24 | 15 |

+----+------+------+------+------+------+------+------+------+------+------+------+------+

In both forms t denotes time, counted usually from midnight, one hour

of time being interpreted as 15° of angle. Form (i) is that utilized

in actually calculating the constants a, b, ... Once the a, b, ...

constants are known, the c, [alpha], ... constants are at once

derivable from the formulae:--

tan [alpha]_n = a_n/b_n;

c_n = a_n/sin [alpha]_n = b_n/cos [alpha]_n = [root](a_n² + b_n²).

The a, b, c, [alpha] constants are called sometimes Fourier, sometimes

Bessel coefficients.

[Illustration: (From _Phil. Trans._)

FIG. 7.]

By taking a sufficient number of terms a series can always be obtained

which will represent any set of diurnal inequality figures; but unless

one can obtain a close approach to the observational figures from the

terms possessing the periods 24, 12, 8 and 6 hours the physical

significance and general utility of the analysis is somewhat

problematical. In the case of the magnetic elements, the 24 and 12

hour terms are usually much the more important; the 24-hour term is

generally, but by no means always, the larger of the two. The c

constants give the amplitudes of the harmonic terms or waves, the

[alpha] constants the phase angles. An advance of 1 hour in the time

of occurrence of the first (and subsequent, if any) maximum and

minimum answers to an _increase_ of 15° in [alpha]1 of 30° in

[alpha]2, of 45° in [alpha]3, of 60° in [alpha]4 and so on. In the

case of magnetic elements the phase angles not infrequently possess a

somewhat large annual variation. It is thus essential for a minute

study of the phenomena at any station to carry out the analysis for

the different seasons of the year, and preferably for the individual

months. If the a and b constants are known for all the individual

months of one year, or for all the Januarys of a series of years, we

have only to take their arithmetic means to obtain the corresponding

constants for the mean diurnal inequality of the year, or for the

diurnal inequality of the average January of the series of years.

This, however, is obviously not true of the c or [alpha] constants,

unless the phase angle is absolutely unchanged throughout the

contributory months or years. This is a point requiring careful

attention, because when giving values of c and [alpha] for the whole

year some authorities give the arithmetic mean of the c's and

[alpha]'s calculated from the diurnal inequalities of the individual

months of the year, others give the values obtained for c and [alpha]

from the mean diurnal inequality of the whole year. The former method

inevitably supplies a larger value for c than the latter, supposing

[alpha] to vary with the season. At some observatories, e.g. Greenwich

and Batavia, it has long been customary to publish every year values

of the Fourier coefficients for each month, and to include other

elements besides the declination. For a thoroughly satisfactory

comparison of different stations, it is necessary to have data from

one and the same epoch; and preferably that epoch should include at

least one 11-year period. There are, however, few stations which can

supply the data required for such a comparison and we have to make the

best of what is available. Information is naturally most copious for

the declination. For this element E. Engelenburg[20] gives values of

C1, C2, C3, C4, and of [alpha]1, [alpha]2, [alpha]3, [alpha]4 for each

month of the year for about 50 stations, ranging from Fort Rae (62° 6´

N. lat.) to Cape Horn (55° 5´ S. lat.). From the results for

individual stations, Engelenburg derives a series of means which he

regards as representative of 11 different zones of latitude. His data

for individual stations refer to different epochs, and some are based

on only one year's observations. The original observations also differ

in reliability; thus the results are of somewhat unequal value. The

mean results for Engelenburg's zones must naturally have some of the

sources of uncertainty reduced; but then the fundamental idea

represented by the arrangement in zones is open to question. The

majority of the data in Table XVII. are taken from Engelenburg, but

the phase angles have been altered so as to apply to westerly

declination. The stations are arranged in order of latitude from north

to south; in a few instances results are given for quiet days. The

figures represent in all cases arithmetic means derived from the 12

monthly values. In the table, so far as is known, the local mean time

of the observatory has been employed. This is a point requiring

attention, because most observatories employ Greenwich time, or time

based on Greenwich or some other national observatory, and any

departure from local time enters into the values of the constants. The

data for Victoria Land refer to the "Discovery's" 1902-1903 winter

quarters, where the declination, taken westerly, was about 207°.5.

As an example of the significance of the phase angles in Table XVII.,

take the ordinary day data for Kew. The times of occurrence of the

maxima are given by t + 234° = 450° for the 24-hour term, 2t + 39°.7 =

90° or = 450° for the 12-hour term, and so on, taking an hour in t as

equivalent to 15°.

Thus the times of the maxima are:--

24-hour term, 2 h. 24 m. p.m.; 12-hour term, 1 h. 41 m. a.m. and p.m.

8-hour term, 4 h. 41 m. a.m., 0 h. 41 m. p.m., and 8 h. 41 m. p.m.

6-hour term, 0 h. 33 m. a.m. and p.m., and 6 h. 33 m. a.m. and p.m.

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.

[Illustration: (_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.

+---------------------+-----------+-------+-------+-------+-------+-----------+-----------+-----------+-----------+

+---------------------+-----------+-------+-------+-------+-------+-----------+-----------+-----------+-----------+

| | | ´ | ´ | ´ | ´ | ° | ° | ° | ° |

| Fort Rae (all) | 1882-1883 | 18.49 | 8.22 | 1.99 | 2.07 | 156.5 | 41.9 | 308 | 104 |

| " (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 |

| " (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 1[gamma] 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. | [alpha]1. | [alpha]2. | [alpha]3. | [alpha]4. |

+----------+-------+-------+-------+-------+-----------+-----------+-----------+-----------+

| | ´ | ´ | ´ | ´ | ° | ° | ° | ° |

| 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 |

+----------+-------+-------+-------+-------+-----------+-----------+-----------+-----------+

Annual Inequality.

§ 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 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,[21] 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[22] 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).

+-------------+-------+-------+-------+-------+-----------+-----------+-----------+-----------+

| Month. | C1. | C2. | C3. | C4. | [alpha]1. | [alpha]2. | [alpha]3. | [alpha]4. |

+-------------+-------+-------+-------+-------+-----------+-----------+-----------+-----------+

| | ´ | ´ | ´ | ´ | ° | ° | ° | ° |

| /Winter | 0.240 | 0.222 | 0.104 | 0.076 | 250.0 | 91.8 | 344 | 194 |

| I < 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 |

| | | | | | | | | |

| /Winter | 3.62 | 3.86 | 1.81 | 1.13 | 82.9 | 277.3 | 154 | 6 |

| H < 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 |

| | | | | | | | | |

| /Winter | 2.46 | 1.67 | 0.86 | 0.42 | 153.9 | 300.8 | 108 | 280 |

| V < 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 |

+-------------+-------+-------+-------+-------+-----------+-----------+-----------+-----------+

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,

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 Lüdeling 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.

TABLE XX.--Annual Inequality.

+---------------------------------------------------------------------------------------------+-------------------------------------+

| Declination. | Inclination. |

+-----------+---------+-----------+-----------+-----------------------+-----------+-----------+---------+---------+---------+-------+

| | sphere. | | 1888-1897.| q. | o. | s. | | | mean. | | | |

+-----------+---------+-----------+-----------+-------+-------+-------+-----------+-----------+---------+---------+---------+-------+

| | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ |

| January | -0.25 | +0.04 | +0.01 | +0.08 | +0.03 | +0.32 | +0.23 | +0.06 | +0.49 | +0.32 | +0.44 | -0.03 |

| February | -0.54 | -0.11 | 0.00 | +0.48 | +0.25 | -0.20 | +0.19 | +0.29 | +0.39 | +0.56 | +0.29 | -0.07 |

| March | -0.27 | +0.04 | +0.17 | +0.03 | +0.05 | -1.02 | -0.12 | +0.27 | +0.20 | +0.38 | +0.13 | +0.53 |

| April | -0.03 | +0.10 | +0.12 | -0.31 | -0.14 | -0.90 | -0.11 | +0.30 | -0.08 | -0.02 | -0.13 | +0.18 |

| May | +0.19 | +0.07 | -0.11 | -0.39 | -0.28 | +0.29 | -0.30 | +0.08 | -0.43 | -0.29 | -0.37 | -0.15 |

| June | +0.46 | +0.13 | -0.14 | -0.47 | -0.39 | +0.78 | -0.13 | -0.19 | -0.70 | -0.77 | -0.59 | -0.35 |

| July | +0.48 | +0.14 | -0.17 | -0.30 | -0.13 | +0.44 | -0.08 | -0.44 | -0.72 | -0.67 | -0.27 | -0.13 |

| August | +0.47 | +0.11 | +0.01 | +0.08 | +0.05 | +0.52 | -0.18 | -0.38 | -0.47 | -0.23 | -0.05 | -0.19 |

| September | +0.31 | +0.01 | 0.00 | +0.29 | +0.24 | -0.02 | +0.06 | -0.06 | -0.06 | +0.16 | +0.01 | +0.20 |

| October | -0.07 | -0.11 | +0.09 | +0.06 | +0.01 | -0.26 | +0.03 | -0.04 | +0.31 | +0.27 | +0.19 | 0.00 |

| November | -0.30 | -0.28 | -0.05 | +0.17 | +0.11 | -0.02 | +0.08 | -0.01 | +0.51 | +0.30 | +0.43 | +0.18 |

| December | -0.36 | -0.14 | +0.05 | +0.26 | +0.23 | +0.05 | +0.35 | +0.06 | +0.55 | +0.19 | +0.24 | -0.29 |

+-----------+---------+-----------+-----------+-------+-------+-------+-----------+-----------+---------+---------+---------+-------+

| Range | 1.02 | 0.42 | 0.34 | 0.95 | 0.64 | 1.80 | 0.65 | 0.74 | 1.27 | 1.33 | 1.03 | 0.88 |

+-----------+---------+-----------+-----------+-------+-------+-------+-----------+-----------+---------+---------+---------+-------+

Annual Variation Fourier Coefficients.

§ 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 submultiples. Fourier series can also be formed

representing the annual variation in the amplitudes of the regular

diurnal inequality, and its component 24-hour, 12-hour, &c. waves, or

of the amplitude of the absolute 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 12 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 + [theta]1) + P2 sin (2t + [theta]2) + ...,

where t is time counted from the beginning of the year, one month

being taken as the equivalent of 30°, P1, P2 represent the amplitudes,

and [theta]1, [theta]2 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.[23a] In the case of P1 and P2 the unit is 1´ for D and I,

and 1[gamma] for H and V. M denotes the mean value of the range for

the 12 months. The letters q and o 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 [theta]1 or [theta]2 means an earlier occurrence of the

maximum or maxima, 1° answering roughly to one day in the case of the

12-month term, and to half a day in the case of the 6-month term. P1/M

and P2/M 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 from

quiet days to ordinary days, or from years of sun spot minimum to

years of sun spot maximum, which appears in the table, appears also

in the case of the horizontal force--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.

TABLE XXI.--Annual Variation of Diurnal Inequality Range. Fourier

Coefficients.

+---------------------+--------+--------+-----------+-----------+-------+-------+

| | P1. | P2. | [theta]1. | [theta]2. | P1/M. | P2/M. |

+-----------+---------+--------+--------+-----------+-----------+-------+-------+

| Kew | D_0 | 3.36 | 0.94 | 279° | 280° | 0.40 | 0.11 |

| 1890-1900 | D_q | 3.81 | 1.22 | 275° | 273° | 0.47 | 0.15 |

| | I_q | 0.67 | 0.16 | 264° | 269° | 0.42 | 0.10 |

| | H_q | 13.6 | 3.0 | 269° | 261° | 0.48 | 0.11 |

| | V_q | 11.7 | 2.2 | 282° | 242° | 0.63 | 0.12 |

+-----------+---------+--------+--------+-----------+-----------+-------+-------+

| S max. | Kew | 4.50 | 1.26 | 277° | 282° | 0.47 | 0.13 |

| D_q | Falmouth| 4.10 | 1.40 | 277° | 286° | 0.43 | 0.15 |

+-----------+---------+--------+--------+-----------+-----------+-------+-------+

| S min. | Kew | 3.35 | 1.10 | 274° | 269° | 0.49 | 0.16 |

| D_q | Falmouth| 3.19 | 1.14 | 275° | 277° | 0.49 | 0.17 |

+-----------+---------+--------+--------+-----------+-----------+-------+-------+

Absolute Range.

§ 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 not vary very much

with the season, but still it does vary. The principal minimum of the

day, for instance, 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 saltum._ 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 different 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 11 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 12 months of the year

(the eleven Januarys being combined into one, 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 suffice 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

range in the mean diurnal inequality for the year, however this

latter is formed. Other causes, moreover, are at work tending in the

same direction. Even in central Europe, the magnetic curves for

individual days of an ordinary month often differ widely amongst

themselves, and show maxima and minima at different times of the day.

In high latitudes, the variation from day to day is sometimes so great

that mere eye inspection of magnetograph curves may leave one with but

little idea as to the probable shape of the resultant diurnal curve

for the month. Table XXII. gives the arithmetic mean of the absolute

daily ranges from a few stations. The values which it assigns to the

year are the arithmetic means of the 12 monthly values. The Mauritius

data are for different periods, viz. declination 1875, 1880 and 1883

to 1890, horizontal force 1883 to 1890, vertical force 1884 to 1890.

The other data are all for the period 1890 to 1900.

TABLE XXII.--Mean Absolute Daily Ranges (Units 1´ for Declination,

1[gamma] for H and V).

+--------------------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+

| | Jan. | Feb. | Mar. | April.| May. | June. | July. | Aug. | Sept. | Oct. | Nov. | Dec. | Year. |

| +-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+

| Pavlovsk | 13.42 | 17.20 | 18.22 | 17.25 | 17.76 | 15.91 | 16.89 | 16.57 | 16.75 | 15.70 | 13.87 | 12.37 | 15.99 |

| Ekatarinburg | 7.33 | 9.54 | 11.90 | 12.89 | 13.63 | 13.03 | 12.78 | 12.21 | 11.23 | 9.44 | 7.86 | 6.85 | 10.72 |

| Kew. All days | 11.16 | 13.69 | 15.93 | 15.00 | 14.90 | 13.65 | 14.13 | 14.22 | 14.57 | 14.07 | 11.71 | 9.80 | 13.57 |

| " Ordinary days| 10.14 | 11.87 | 14.19 | 14.24 | 13.85 | 13.26 | 13.47 | 13.67 | 13.71 | 13.10 | 10.40 | 9.00 | 12.58 |

| " Quiet " | 6.12 | 7.57 | 10.59 | 11.84 | 12.09 | 11.95 | 11.60 | 11.93 | 10.86 | 9.16 | 6.54 | 5.08 | 9.61 |

| Zi-ka-wei | 3.88 | 3.25 | 6.22 | 7.04 | 7.15 | 7.40 | 7.77 | 8.06 | 6.73 | 4.68 | 2.91 | 2.52 | 5.63 |

| Mauritius | 6.93 | 7.79 | 7.11 | 5.75 | 4.87 | 4.03 | 4.36 | 6.00 | 6.28 | 6.71 | 6.99 | 6.78 | 6.13 |

| | | | | | | | | | | | | | |

| Pavlovsk | 52.4 | 74.5 | 79.1 | 80.1 | 86.2 | 79.0 | 86.7 | 77.6 | 76.7 | 67.3 | 55.7 | 45.9 | 71.8 |

| Ekatarinburg | 33.2 | 43.1 | 48.4 | 51.7 | 56.2 | 54.1 | 56.7 | 51.7 | 49.3 | 44.1 | 34.1 | 29.3 | 46.0 |

| Mauritius | 37.9 | 35.0 | 36.2 | 37.6 | 35.0 | 34.1 | 33.8 | 34.5 | 36.6 | 37.4 | 37.8 | 35.3 | 35.9 |

| | | | | | | | | | | | | | |

| Pavlovsk | 27.0 | 50.4 | 54.7 | 43.2 | 45.3 | 34.8 | 42.1 | 35.5 | 42.5 | 37.5 | 33.5 | 25.5 | 39.3 |

| Ekatarinburg | 17.4 | 26.6 | 29.2 | 30.1 | 29.6 | 27.6 | 29.6 | 26.1 | 25.2 | 22.1 | 19.6 | 16.4 | 24.9 |

| Mauritius | 17.1 | 19.5 | 20.1 | 17.3 | 16.5 | 15.5 | 17.1 | 22.0 |22.7 | 19.4 | 16.7 | 15.2 | 18.2 |

+--------------------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+

A comparison of the absolute ranges in Table XXII. with the inequality

ranges for the same stations derivable from Tables VIII. to X. is most

instructive. At Mauritius the ratio of the absolute to the inequality

range is for D 1.38, for H 1.76, and for V 1.19. At Pavlovsk the

corresponding ratios are much larger, viz. 2.16 for D, 2.43 for H, and

2.05 for V. The declination data for Kew in Table XXII. illustrate

other points. The first set of data are derived from all days of the

year. The second omit the highly disturbed days. The third answer to

the 5 days a month selected as typically quiet. The yearly mean

absolute range from ordinary days at Kew in Table XXII. is 1.49 times

the mean inequality range in Table VIII.; comparing individual months

the ratio of the absolute to the inequality range varies from 2.06 in

January to 1.21 in June. Even confining ourselves to the quiet days at

Kew, which are free from any but the most trifling disturbances, we

find that the mean absolute range for the year is 1.20 times the

arithmetic mean of the inequality ranges for the individual months of

the year, and 1.22 times the range from the mean diurnal inequality

for the year. In this case the ratio of the absolute to the inequality

range varies from 1.55 in December to only 1.09 in May.

§ 25. The variability of the absolute daily range of declination is

illustrated by Table XXIII., which contains data for Kew[24] derived

from all days of the 11-year period 1890-1900. It gives the total

number of times during the 11 years when the absolute range lay within

the limits specified at the heads of the first nine columns of

figures. The two remaining columns give the arithmetic means of the

five largest and the five least absolute ranges encountered each

month. The mean of the twelve monthly diurnal inequality ranges from

ordinary days was only 8´.44, but the absolute range during the 11

years exceeded 20´ on 492 days, 15´ on 1196 days, and 10´ on 2784

days, i.e. on 69 days out of every 100.

Table XXIII.--Absolute Daily Range of Declination at Kew.

+------------------------------------------------------------------------------------------------------------------+---------------------+

| | Means from the 5 |

| | largest and 5 least |

| Number of occasions during 11 years when absolute range was:-- | 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 | 226 | 10 | 4 | 2 | 8 | 27.21 | 6.55 |

| March | 1 | 72 | 138 | 61 | 232 | 21 | 8 | 1 | 7 | 29.87 | 8.93 |

| April | 0 | 43 | 167 | 73 | 227 | 10 | 6 | 3 | 1 | 23.69 | 10.31 |

| May | 0 | 57 | 157 | 85 | 220 | 12 | 3 | 0 | 7 | 25.36 | 9.50 |

| June | 0 | 56 | 185 | 67 | 215 | 1 | 3 | 1 | 2 | 19.92 | 9.89 |

| July | 0 | 59 | 185 | 70 | 214 | 5 | 2 | 2 | 4 | 22.49 | 9.96 |

| August | 0 | 37 | 202 | 75 | 222 | 1 | 2 | 0 | 2 | 21.27 | 10.05 |

| September | 1 | 68 | 153 | 71 | 219 | 5 | 4 | 5 | 4 | 24.55 | 9.52 |

| October | 3 | 103 | 111 | 67 | 234 | 10 | 11 | 2 | 0 | 23.92 | 8.01 |

| November | 42 | 140 | 81 | 28 | 214 | 9 | 8 | 5 | 3 | 23.58 | 5.64 |

| December | 64 | 166 | 56 | 29 | 214 | 7 | 1 | 1 | 3 | 20.43 | 4.36 |

+-----------+---------+----------+-----------+-----------+-----------+-----------+-----------+-----------+---------+-----------+---------+

| Totals | 188 | 1045 | 1588 | 714 | 261 | 98 | 56 | 25 | 42 | | |

+-----------+---------+----------+-----------+-----------+-----------+-----------+-----------+-----------+---------+-----------+---------+

Relations to Sun-spot Frequency.

§ 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 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.

+----------+-------------------------+-----------------------------------+---------------------------------+

+----------+-------+-------+---------+-------+-------+---------+---------+-------+-------+---------+-------+

| | D. | I. | H. | D. | I. | H. | V. | D_q. | I_q. | H_q. | D_0. |

| +-------+-------+---------+-------+-------+---------+---------+-------+-------+---------+-------+

| | ´ | ´ | [gamma] | ´ | ´ | [gamma] | [gamma] | ´ | ´ | [gamma] | ´ |

| 1890_11 | 6.32 | 1.33 | 22 | 5.83 | 1.05 | 18 | 9 | 6.90 | | 20 | 7.32 |

| 1891_6 | 7.31 | 1.79 | 30 | 6.85 | 1.38 | 25 | 14 | 8.04 | 1.52 | 28 | 8.48 |

| 1892_3 | 8.75 | 2.21 | 37 | 7.74 | 1.72 | 32 | 19 | 9.50 | 1.66 | 31 | 9.85 |

| 1893_1 | 9.64 | 2.24 | 38 | 8.83 | 1.80 | 31 | 17 | 10.06 | 1.96 | 35 | 10.74 |

| 1894_2 | 8.58 | 2.17 | 38 | 7.80 | 1.73 | 30 | 17 | 9.32 | 1.94 | 34 | 9.80 |

| 1895_4 | 8.22 | 2.08 | 33 | 7.29 | 1.64 | 28 | 15 | 8.59 | 1.66 | 30 | 9.54 |

| 1896_5 | 7.39 | 1.77 | 29 | 6.50 | 1.38 | 25 | 15 | 7.77 | 1.31 | 25 | 8.50 |

| 1897_6 | 6.79 | 1.59 | 26 | 6.01 | 1.16 | 21 | 12 | 6.71 | 1.14 | 22 | 7.76 |

| 1898_7 | 6.25 | 1.56 | 26 | 5.76 | 1.19 | 21 | 11 | 6.85 | 1.07 | 21 | 7.59 |

| 1899_9 | 6.02 | 1.44 | 24 | 5.33 | 1.12 | 20 | 11 | 6.69 | 1.01 | 21 | 7.30 |

| 1900_10 | 6.20 | 1.28 | 22 | 5.88 | 0.93 | 17 | 8 | 6.52 | 1.06 | 21 | 6.83 |

+----------+-------+-------+---------+-------+-------+---------+---------+-------+-------+---------+-------+

TABLE XXV.--Absolute Ranges.

+---------+-----------------------+-----------------------------------------------------------------------------------+

| | Kew Declination. | Pavlovsk. |

| | Daily. +---------------------------+---------------------------+---------------------------+

+---------+-------+-------+-------+-------+---------+---------+-------+---------+---------+-------+---------+---------+

| | q. | o. | a. | D. | H. | V. | D. | H. | V. | D. | H. | V. |

| +-------+-------+-------+-------+---------+---------+-------+---------+---------+-------+---------+---------+

| | ´ | ´ | ´ | ´ | [gamma] | [gamma] | ´ | [gamma] | [gamma] | ´ | [gamma] | [gamma] |

| 1890_11 | 8.3 | 10.5 | 10.7 | 12.1 | 49 | 21 | 28.2 | 118 | 80 | 42.1 | 169 | 179 |

| 1891_6 | 10.0 | 12.8 | 13.7 | 16.0 | 70 | 39 | 46.3 | 218 | 233 | 92.3 | 550 | 614 |

| 1892_3 | 12.3 | 15.4 | 17.7 | 21.0 | 111 | 73 | 93.6 | 698 | 575 | 194.0 | 2416 | 1385 |

| 1893_1 | 11.8 | 15.2 | 15.6 | 17.8 | 79 | 41 | 48.3 | 241 | 210 | 87.1 | 514 | 457 |

| 1894_2 | 11.3 | 14.7 | 16.5 | 20.4 | 97 | 62 | 84.1 | 493 | 493 | 145.6 | 1227 | 878 |

| 1895_4 | 10.6 | 14.8 | 15.6 | 18.1 | 80 | 46 | 47.4 | 220 | 223 | 73.9 | 395 | 534 |

| 1896_5 | 9.5 | 12.9 | 14.5 | 17.5 | 74 | 43 | 52.4 | 232 | 236 | 88.7 | 574 | 608 |

| 1897_8 | 8.2 | 11.5 | 12.1 | 14.6 | 61 | 30 | 43.8 | 201 | 170 | 101.1 | 449 | 480 |

| 1898_7 | 8.2 | 11.2 | 12.3 | 14.7 | 67 | 35 | 46.6 | 276 | 242 | 118.9 | 1136 | 888 |

| 1899_9 | 7.9 | 10.5 | 11.3 | 13.1 | 58 | 27 | 38.3 | 178 | 150 | 63.8 | 382 | 527 |

| 1900_10 | 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 [equiv] 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.

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[25] 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. |--------+-------+ Epoch. +--------+-------+---------+--------+

| | 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 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 Wolf'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.

TABLE XXVII.--Values of a and b in Wolf's Formula.

+---------------------------------+----------------------+----------------------+----------------------+----------------------+

+---------------------------------+------+------+--------+------+------+--------+------+------+--------+------+------+--------+

| Diurnal Inequality for the Year.| a. | b. |100 b/a.| a. | b. |100 b/a.| a. | b. |100 b/a.| a. | b. |100 b/a.|

+---------------------------------+------+------+--------+------+------+--------+------+------+--------+------+------+--------+

| Pavlovsk, 1890-1900 all | 5.74 |.0400 | .70 | 1.24 |.0126 | 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 | .117 | 1.37 |

| Irkutsk " " all | 4.82 |.0358 | .74 | 0.97 |.0087 | 0.90 | 18.2 | .190 | 1.04 | 6.5 | .071 | 1.09 |

| Kew " " quiet | 6.10 |.0433 | .71 | 0.87 |.0125 | 1.45 | 18.1 | .194 | 1.07 | 14.3 | .081 | 0.56 |

| Falmouth, 1891-1902 quiet | 5.90 |.0451 | .76 | .. | .. | .. | 20.1 | .233 | 1.16 | .. | .. | .. |

| 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 / 1875-1880 \ all | 4.06 |.0164 | .40 | .. | .. | .. | 15.0 | .096 | 0.64 | 11.9 | .069 | 0.58 |

| \ 1883-1890 / | | | | | | | | | | | | |

+---------------------------------+------+------+--------+------+------+--------+------+------+--------+------+------+--------+

| 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 | 1.06 | 19.2 | .195 | 1.01 | 9.2 | .156 | 1.70 |

| 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 |.0130 | 1.11 | 21.5 | .191 | 0.89 | 16.0 | .072 | 0.45 |

| Falmouth, 1891-1902 quiet | 6.16 |.0450 | .73 | .. | .. | .. | 20.9 | .236 | 1.13 | .. | .. | .. |

+---------------------------------+------+------+--------+------+------+--------+------+------+--------+------+------+--------+

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

0.81 in winter to 0.52 in summer on all days, but from 1.39 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

1.77, summer 0.98, but for quiet days--winter 1.83, summer 0.71.

Wolf'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 inequality ranges, but the accordance between observed values

and those calculated from Wolf'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 sun-spots.

§ 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 sun-spots 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-spot maximum)

denotes mean average results from the four years 1892-1895, 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 12-hour 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, i.e. 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 15.6 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.

+---------+-------------+-------------+-------------+-------------+

| +------+------+------+------+------+------+------+------+

+---------+------+------+------+------+------+------+------+------+

| | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ |

| c1 | 3.47 | 2.21 | 2.41 | 1.43 | 3.76 | 2.41 | 4.38 | 2.98 |

| c2 | 2.04 | 1.51 | 1.15 | 0.78 | 2.33 | 1.71 | 2.73 | 2.06 |

| c3 | 0.89 | 0.72 | 0.55 | 0.42 | 1.16 | 0.97 | 0.97 | 0.77 |

| c4 | 0.28 | 0.27 | 0.30 | 0.27 | 0.42 | 0.42 | 0.11 | 0.11 |

+---------+------+------+------+------+------+------+------+------+

| | ° | ° | ° | ° | ° | ° | ° | ° |

|[alpha]1 | 228.5| 232.4| 243.0| 256.0| 231.3| 233.7| 218.2| 220.3|

|[alpha]2 | 41.7| 46.6| 23.5| 36.9| 40.6| 43.9| 50.6| 52.5|

|[alpha]3 | 232.6| 243.6| 234.0| 257.6| 228.4| 236.2| 236.8| 245.4|

|[alpha]4 | 58.0| 57.3| 52.3| 60.8| 62.0| 58.2| 57.4| 45.2|

+---------+------+------+------+------+------+------+------+------+

Quiet Day Phenomena.

§ 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 in

the absence of irregular disturbances, and that mean annual values, or

secular change data, or diurnal inequalities, derived from them might

be regarded as truly normal or representative of the station. It was

found, however, by P. A. Müller[29] 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[30] at Greenwich for the period 1889-1896. The average excesses

for the quiet-day over the all-day means in these two cases were as

follows:--

+---------------+--------------+--------------+------------+-------------+

+---------------+--------------+--------------+------------+-------------+

+---------------+--------------+--------------+------------+-------------+

The sign of the difference in the case of D, I and H was the 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 small

and might be accidental. In the case of D at Greenwich 1891 differed

from the other years, and of two more recent years examined by

Ellis[31] one, 1904, agreed with 1891. At Kew, on the average of the

11 years 1890 to 1900, the quiet-day mean annual value of declination

exceeded the ordinary day value, but the apparent excess 0´.02 is too

small to possess much significance.

Non-cyclic Change.

Another property more recently discovered in quiet days is the

non-cyclic change. The nature of this phenomenon will be readily

understood from the following data from the 11-year period 1890 to

1900 at Kew[32]. The mean daily change for all days is calculated from

the observed annual change.

+------------------------------+--------+--------+--------------+--------------+

| | D. | I. | H. | V. |

+------------------------------+--------+--------+--------------+--------------+

| | ´ | ´ | | |

+------------------------------+------------------+-------------+--------------+

Thus the changes during the representative quiet day differed from

those of the average day. Before accepting such a phenomenon as

natural, instrumental peculiarities must be carefully considered. The

secular change is really based on the absolute instruments, the

diurnal changes on the magnetographs, and the first idea likely to

occur to a critical mind is that the apparent abnormal change on quiet

days represents in reality change of zero in the magnetographs. If,

however, the phenomenon were instrumental, it should appear equally on

days other than quiet days, and we should thus have a shift of zero

amounting in a year to over 1,200[gamma] in H, and to about 90´ in I.

Under such circumstances the curve would be continually drifting off

the sheet. In the case of the Kew magnetographs, a careful

investigation showed that if any instrumental change occurred in the

declination magnetograph during the 11 years it did not exceed a few

tenths of a minute. In the case of the H and V magnetographs at Kew

there is a slight drift, of instrumental origin, due to weakening of

the magnets, but it is exceedingly small, and in the case of H is in

the opposite direction to the non-cyclic change on quiet days. It only

remains to add that the hypothesis of instrumental origin was

positively disproved by measurement of the curves on ordinary days.

It must not be supposed that every quiet day agrees with the average

quiet day in the order of magnitude, or even in the sign, of the

non-cyclic change. In fact, in not a few months the sign of the

non-cyclic change on the mean of the quiet days differs from that

obtained for the average quiet day of a period of years. At Kew,

between 1890 and 1900, the number of months during which the mean

non-cyclic change for the five quiet days selected by the astronomer

royal (Sir W. H. M. Christie) was plus, zero, or minus, was as

follows:--

+-----------+------+------+------+------+

| Element. | D. | I. | H. | V. |

+-----------+------+------+------+------+

| Number + | 63 | 13 | 112 | 47 |

| " 0 | 14 | 16 | 11 | 9 |

| " - | 55 | 101 | 9 | 74 |

+-----------+------+------+------+------+

The + sign denotes westerly movement in the declination, and

increasing dip of the north end of the needle. In the case of I and H

the excess in the number of months showing the normal sign is

overwhelming. The following mean non-cyclic changes on quiet days are

from other sources:--

+-----------+--------------+-------------+--------------+

| Element. | (1890-1895). | (1898-1902).| (1894-1901). |

+-----------+--------------+-------------+--------------+

| | ´ | ´ | ´ |

| D | + 0.03 | + 0.05 | + 0.07 |

+-----------+--------------+-------------+--------------+

The results are in the same direction as at Kew, + meaning in the case

of D movement to the west. At Falmouth[32], as at Kew, the non-cyclic

change showed a tendency to be small in years of few sun-spots.

§ 30. In calculating diurnal inequalities from quiet days the

non-cyclic effect must be eliminated, otherwise the result would

depend on the hour at which the "day" is supposed to commence. If the

value recorded at the second midnight of the average day exceeds that

at the first midnight by N, the elimination is effected by applying to

each hourly value the correction N(12 - n)/24, where n is the hour

counted from the first midnight (0 hours). This assumes the change to

progress uniformly throughout the 24 hours. Unless this is practically

the case--a matter difficult either to prove or disprove--the

correction may not secure exactly what is aimed at. This method has

been employed in the previous tables. The fact that differences do

exist between diurnal inequalities derived from quiet days and all

ordinary days was stated explicitly in § 4, and is obvious in Tables

VIII. to XI. An extreme case is represented by the data for Jan Mayen

in these tables. Figs. 9 and 10 are vector diagrams for this station,

for all and for quiet days during May, June and July 1883, according

to data got out by Lüdeling. As shown by the arrows, fig. 10 (quiet

days) is in the main described in the normal or clockwise direction,

but fig. 9 (all days) is described in the opposite direction. Lüdeling

found this peculiar difference between all and quiet days at all the

north polar stations occupied in 1882-1883 except Kingua Fjord, where

both diagrams were described clockwise.

[Illustration: FIG. 9.]

[Illustration: FIG. 10.]

In temperate latitudes the differences of type are much less, but

still they exist. A good idea of their ordinary size and character in

the case of declination may be derived from Table XXIX., containing

data for Kew, Greenwich and Parc St Maur.

The data for Greenwich are due to W. Ellis[30], those for Parc St Maur

to T. Moureaux[33]. The quantity tabulated is the algebraic excess of

the all or ordinary day mean hourly value over the corresponding quiet

day value in the mean diurnal inequality for the year. At Greenwich

and Kew days of extreme disturbance have been excluded from the

ordinary days, but apparently not at Parc St Maur. The number of

highly disturbed days at the three stations is, however, small, and

their influence is not great. The differences disclosed by Table XXIX.

are obviously of a systematic character, which would not tend to

disappear however long a period was utilized. In short, while the

diurnal inequality from quiet days may be that most truly

representative of undisturbed conditions, it does not represent the

average state of conditions at the station. To go into full details

respecting the differences between all and quiet days would occupy

undue space, so the following brief summary of the differences

observed in declination at Kew must suffice. While the inequality

range is but little different for the two types of days, the mean of

the hourly differences from the mean for the day is considerably

reduced in the quiet days. The 24-hour term in the Fourier analysis is

of smaller amplitude in the quiet days, and its phase angle is on the

average about 6°.75 smaller than on ordinary days, implying a

retardation of about 27 minutes in the time of maximum. The diurnal

inequality range is more variable throughout the year in quiet days

than on ordinary days, and the same is true of the absolute ranges.

The tendency to a secondary minimum in the range at midsummer is

considerably more decided on ordinary than on quiet days. When the

variation throughout the year in the diurnal inequality range is

expressed in Fourier series, whose periods are the year and its

submultiples, the 6-month term is notably larger for ordinary than for

quiet days. Also the date of the maximum in the 12-month term is about

three days earlier for ordinary than for quiet days. The exact size of

the differences between ordinary and quiet day phenomena must depend

to some extent on the criteria employed in selecting quiet days and in

excluding disturbed days. This raises difficulties when it comes to

comparing results at different stations. For stations near together

the difficulty is trifling. The astronomer royal's quiet days have

been used for instance at Parc St. Maur, Val Joyeux, Falmouth and Kew,

as well as at Greenwich. But when stations are wide apart there are

two obvious difficulties: first, the difference of local time;

secondly, the fact that a day may be typically quiet at one station

but appreciably disturbed at the other.

If the typical quiet day were simply the antithesis of a disturbed

day, it would be natural to regard the non-cyclic change on quiet days

as a species of recoil from some effect of disturbance. This view

derives support from the fact, pointed out long ago by Sabine[34],

that the horizontal force usually, though by no means always, is

lowered by magnetic disturbances. Dr van Bemmelen[35] who has examined

non-cyclic phenomena at a number of stations, seems disposed to regard

this as a sufficient explanation. There are, however, difficulties in

accepting this view. Thus, whilst the non-cyclic effect in horizontal

force and inclination at Kew and Falmouth appeared on the whole

enhanced in years of sun-spot maximum, the difference between years

such as 1892 and 1894 on the one hand, and 1890 and 1900 on the other,

was by no means proportional to the excess of disturbance in the

former years. Again, when the average non-cyclic change of declination

was calculated at Kew for 207 days, selected as those of most marked

irregular disturbance between 1890 and 1900, the sign actually proved

to be the same as for the average quiet day of the period.

TABLE XXIX.--All or Ordinary, less Quiet Day Hourly Values (+ to the

West).

+-----+--------------------------------------+-------------------------------------+

| | Forenoon. | Afternoon. |

|Hour.+------------+-----------+-------------+-----------+-----------+-------------+

| | 1890-1900. | 1890-1894.| 1883-1897. | 1890-1900.| 1890-1894.| 1893-1897. |

+-----+------------+-----------+-------------+-----------+-----------+-------------+

| | ´ | ´ | ´ | ´ | ´ | ´ |

| 1 | -0.58 | -0.59 | -0.63 | +0.42 | +0.44 | +0.40 |

| 2 | -0.54 | -0.47 | -0.47 | +0.52 | +0.45 | +0.50 |

| 3 | -0.51 | -0.31 | -0.32 | +0.57 | +0.52 | +0.59 |

| 4 | -0.41 | -0.23 | -0.16 | +0.60 | +0.51 | +0.55 |

| 5 | -0.28 | -0.10 | -0.01 | +0.46 | +0.34 | +0.38 |

| 6 | -0.08 | +0.12 | +0.18 | +0.21 | +0.04 | +0.07 |

| 7 | +0.13 | +0.30 | +0.34 | -0.06 | -0.24 | -0.25 |

| 8 | +0.29 | +0.48 | +0.47 | -0.27 | -0.50 | -0.54 |

| 9 | +0.40 | +0.56 | +0.53 | -0.47 | -0.68 | -0.74 |

| 10 | +0.44 | +0.58 | +0.51 | -0.61 | -0.78 | -0.79 |

| 11 | +0.48 | +0.50 | +0.44 | -0.62 | -0.77 | -0.79 |

| 12 | +0.45 | +0.44 | +0.38 | -0.54 | -0.61 | -0.67 |

+-----+------------+-----------+-------------+-----------+-----------+-------------+

Magnetic Disturbances.

§ 31. A satisfactory definition of magnetic disturbance is about as

difficult to lay down as one of heterodoxy. The idea in its generality

seems to present no difficulty, but it is a very different matter when

one comes to details. Amongst the chief disturbances recorded since

1890 are those of February 13-14 and August 12, 1892; July 20 and

August 20, 1894; March 15-16, and September 9, 1898; October 31, 1903;

February 9-10, 1907; September 11-12, 1908 and September 25, 1909. On

such days as these the oscillations shown by the magnetic curves are

large and rapid, aurora is nearly always visible in temperate

latitudes, earth currents are prominent, and there is

interruption--sometimes very serious--in the transmission of telegraph

messages both in overhead and underground wires. At the other end of

the scale are days on which the magnetic curves show practically no

movement beyond the slow regular progression of the regular diurnal

inequality. But between these two extremes there are an infinite

variety of intermediate cases. The first serious attempt at a precise

definition of disturbance seems due to General Sabine[35a]. His method

had once an extensive vogue, and still continues to be applied at some

important observatories. Sabine regarded a particular observation as

disturbed when it differed from the mean of the observations at that

hour for the whole month by not less than a certain limiting value.

His definition takes account only of the extent of the departure from

the mean, whether the curve is smooth at the time or violently

oscillating makes no difference. In dealing with a particular station

Sabine laid down separate limiting values for each element. These

limits were the same, irrespective of the season of the year or of the

sun-spot frequency. A departure, for example, of 3´.3 at Kew from the

mean value of declination for the hour constituted a disturbance,

whether it occurred in December in a year of sun-spot minimum, or in

June in a year of sun-spot maximum, though the regular diurnal

inequality range might be four times as large in the second case as in

the first. The limiting values varied from station to station, the

size depending apparently on several considerations not very clearly

defined. Sabine subdivided the disturbances in each element into two

classes: the one tending to increase the element, the other tending to

diminish it. He investigated how the numbers of the two classes varied

throughout the day and from month to month. He also took account of

the aggregate value of the disturbances of one sign, and traced the

diurnal and annual variations in these aggregate values. He thus got

two sets of diurnal variations and two sets of annual variations of

disturbance, the one set depending only on the number of the disturbed

hours, the other set considering only the aggregate value of the

disturbances. Generally the two species of disturbance variations were

on the whole fairly similar. The aggregates of the + and -

disturbances for a particular hour of the day were seldom equal, and

thus after the removal of the disturbed values the mean value of the

element for that hour was generally altered. Sabine's complete scheme

supposed that after the criterion was first applied, the hourly means

would be recalculated from the undisturbed values and the criterion

applied again, and that this process would be repeated until the

disturbed observations all differed by not less than the accepted

limiting value from the final mean based on undisturbed values alone.

If the disturbance limit were so small that the disturbed readings

formed a considerable fraction of the whole number, the complete

execution of Sabine's scheme would be exceedingly laborious. As a

matter of fact, his disturbed readings were usually of the order of 5%

of the total number, and unless in the case of exceptionally large

magnetic storms it is of little consequence whether the first choice

of disturbed readings is accepted as final or is reconsidered in the

light of the recalculated hourly means.

Sabine applied his method to the data obtained during the decade 1840

to 1850 at Toronto, St Helena, Cape of Good Hope and Hobart, also to

data for Pekin, Nertchinsk, Point Barrow, Port Kennedy and Kew. C.

Chambers[36] applied it to data from Bombay. The yearly publication of

the Batavia observatory gives corresponding results for that station,

and Th. Moureaux [33] has published similar data for Parc St Maur.

Tables XXX. to XXXII. are based on a selection of these data. Tables

XXX. and XXXI. show the annual variation in Sabine's disturbances, the

monthly values being expressed as percentages of the arithmetic mean

value for the 12 months. The Parc St Maur and Batavia data, owing to

the long periods included, are especially noteworthy. Table XXX. deals

with the east (E) and west (W) disturbances of declination separately.

Table XXXI., dealing with disturbances in horizontal and vertical

force, combines the + and - disturbances, treated numerically. At Parc

St Maur the limits required to qualify for disturbance were 3´.0 in D,

20[gamma] in H, and 12[gamma] in V; the corresponding limits for

Batavia were 1´.3, 11[gamma] and 11[gamma]. The limits for D at

Toronto, Bombay and Hobart were respectively 3´.6, 1´.4 and 2´.4.

At Parc St Maur the disturbance data from all three elements give

distinct maxima near the equinoxes; a minimum at midwinter is clearly

shown, and also one at midsummer, at least in D and H. A decline in

disturbance at midwinter is visible at all the stations, but at

Batavia the equinoctial values for D and V are inferior to those at

midsummer.

TABLE XXX.--Annual Variation of Disturbances (Sabine's numbers).

+-----------+-------------+-------------+-------------+-------------+-------------+

| | 1883-97. | 1841-48. | 1859-65. | 1883-99. | 1843-48. |

+-----------+------+------+------+------+------+------+------+------+------+------+

| Month. | E. | W. | E. | W. | E. | W. | E. | W. | E. | W. |

+-----------+------+------+------+------+------+------+------+------+------+------+

| January | 78 | 60 | 55 | 66 | 89 | 89 | 180 | 223 | 165 | 182 |

| February | 116 | 92 | 75 | 86 | 94 | 67 | 138 | 144 | 121 | 116 |

| March | 126 | 107 | 92 | 94 | 129 | 97 | 102 | 87 | 114 | 104 |

| April | 105 | 113 | 115 | 114 | 106 | 129 | 67 | 73 | 110 | 102 |

| May | 101 | 118 | 101 | 101 | 63 | 99 | 72 | 71 | 62 | 53 |

| June | 77 | 89 | 95 | 72 | 78 | 81 | 45 | 27 | 32 | 37 |

| July | 82 | 104 | 140 | 126 | 121 | 173 | 62 | 46 | 50 | 49 |

| August | 88 | 113 | 137 | 133 | 154 | 131 | 69 | 69 | 86 | 78 |

| September | 134 | 137 | 163 | 139 | 111 | 108 | 135 | 144 | 135 | 114 |

| October | 119 | 115 | 101 | 111 | 140 | 128 | 95 | 88 | 124 | 123 |

| November | 99 | 94 | 73 | 85 | 43 | 43 | 106 | 91 | 79 | 111 |

| December | 75 | 58 | 51 | 72 | 72 | 55 | 124 | 137 | 123 | 130 |

+-----------+------+------+------+------+------+------+------+------+------+------+

Table XXXII. shows in some cases a most conspicuous diurnal variation

in Sabine's disturbances. The data are percentages of the totals for

the whole 24 hours. But whilst at Batavia the easterly and westerly

disturbances in D vary similarly, at Parc St Maur they follow opposite

laws, the easterly showing a prominent maximum near noon, the westerly

a still more prominent maximum near midnight. The figures in the

second last line of the table, if divided by 0.24, will give the

percentage of hours which show the species of disturbance indicated.

For instance, at Parc St Maur, out of 100 hours, 3 show disturbances

to the west and 3.7 to the east; or in all 6.7 show disturbances of

declination. The last line gives the average size of a disturbance of

each type, the unit being 1´ in D and 1[gamma] in H and V.

TABLE XXXI.--Annual Variation of Disturbances.

+-----------+-------------+-------------+---------------------------+

+-----------+-------------+-------------+-------------+-------------+

+-----------+------+------+------+------+------+------+------+------+

| | H. | V. | H. | V. | H. | V. | H. | V. |

| +------+------+------+------+------+------+------+------+

| January | 81 | 51 | 58 | 56 | 96 | 151 | 89 | 154 |

| February | 96 | 133 | 94 | 74 | 105 | 123 | 110 | 125 |

| March | 126 | 118 | 94 | 108 | 116 | 105 | 117 | 103 |

| April | 94 | 111 | 150 | 149 | 104 | 76 | 105 | 73 |

| May | 108 | 133 | 90 | 112 | 101 | 92 | 105 | 95 |

| June | 90 | 85 | 36 | 50 | 82 | 69 | 79 | 66 |

| July | 99 | 128 | 61 | 71 | 90 | 83 | 95 | 81 |

| August | 113 | 92 | 75 | 108 | 91 | 91 | 98 | 91 |

| September | 119 | 122 | 171 | 160 | 113 | 111 | 114 | 115 |

| October | 101 | 94 | 148 | 129 | 114 | 89 | 104 | 86 |

| November | 104 | 81 | 98 | 75 | 99 | 102 | 100 | 101 |

| December | 70 | 51 | 128 | 100 | 89 | 108 | 84 | 110 |

+-----------+------+------+------+------+------+------+------+------+

At Batavia disturbances increasing and decreasing the element are

about equally numerous, but this is exceptional. Easterly disturbances

of declination predominated at Toronto, Point Barrow, Fort Kennedy,

Kew, Parc St Maur, Bombay and the Falkland Islands whilst the reverse

was true of St Helena, Cape of Good Hope, Pekin and Hobart. At Kew and

Parc St Maur the ratios borne by the eastern to the western

disturbances were 1.19 and 1.23 respectively, and so not much in

excess of unity; but the preponderance of easterly disturbances at the

North American[37] stations was considerably larger than this.

TABLE XXXII.--Diurnal Variation of Disturbances (Sabine's numbers).

+-------------+-----------------------------------------+-----------------------------------------+

| | Parc St Maur. | Batavia. |

| +-------------+-------------+-------------+-------------+-------------+-------------+

| Hour. | D. | H. | V. | D. | H. | V. |

| +------+------+------+------+------+------+------+------+------+------+------+------+

| | E. | W. | + | - | + | - | E. | W. | + | - | + | - |

+-------------+------+------+------+------+------+------+------+------+------+------+------+------+

| 0-3 | 10.1 | 20.3 | 9.0 | 8.3 | 5.7 | |9.2 | 1.1 | 5.8 | 13.1 | 6.6 | 4.0 | 7.4 |

| 3-6 | 12.3 | 8.2 | 8.4 | 8.0 | 6.4 | 10.4 | 7.6 | 7.3 | 14.2 | 4.8 | 6.3 | 10.0 |

| 6-9 | 15.7 | 3.8 | 14.1 | 12.5 | 7.2 | 9.0 | 24.9 | 16.8 | 12.1 | 9.9 | 21.2 | 21.7 |

| 9-noon | 16.2 | 5.1 | 18.0 | 15.6 | 12.9 | 15.4 | 38.5 | 33.0 | 8.6 | 15.8 | 19.8 | 16.4 |

| noon-3 | 19.3 | 6.7 | 15.3 | 16.5 | 18.2 | 18.3 | 18.8 | 24.7 | 16.8 | 21.1 | 23.5 | 22.1 |

| 3-6 | 14.8 | 9.7 | 12.5 | 15.4 | 22.9 | 21.8 | 6.4 | 5.4 | 13.3 | 16.9 | 12.6 | 12.7 |

| 6-9 | 5.7 | 21.2 | 11.4 | 13.2 | 18.9 | 11.2 | 2.3 | 3.4 | 9.9 | 13.6 | 7.1 | 4.1 |

| 9-12 | 5.9 | 25.0 | 11.2 | 10.5 | 7.8 | 4.7 | 0.4 | 3.8 | 12.0 | 11.1 | 5.6 | 5.4 |

+-------------+------+------+------+------+------+------+------+------+------+------+------+------+

| per day | 0.88| 0.72| 1.15| 1.56| 1.04| 0.96| 0.46| 0.44| 1.62| 1.61| 1.19| 1.13|

| Mean size | .. | .. | .. | .. | .. | .. | 1.72| 1.69| 18.0| 19.5 | 16.7 | 15.5 |

+-------------+------+------+------+------+------+------+------+------+------+------+------+------+

§ 32. From the point of view of the surveyor there is a good deal to

be said for Sabine's definition of disturbance, but it is less

satisfactory from other standpoints. One objection has been already

indicated, viz. the arbitrariness of applying the same limiting value

at a station irrespective of the size of the normal diurnal range at

the time. Similarly it is arbitrary to apply the same limit between 10

a.m. and noon, when the regular diurnal variation is most rapid, as

between 10 p.m. and midnight, when it is hardly appreciable. There

seems a distinct difference of phase between the diurnal inequalities

on different types of days at the same season; also the phase angles

in the Fourier terms vary continuously throughout the year, and much

more rapidly at some stations and at some seasons than at others. Thus

there may be a variety of phenomena which one would hesitate to regard

as disturbances which contribute to the annual and diurnal variations

in Tables XXX. to XXXII.

Sabine, as we have seen, confined his attention to the departure of

the hourly reading from the mean for that hour. Another and equally

natural criterion is the apparent character of the magnetograph curve.

At Potsdam curves are regarded as "1" quiet, "2" moderately disturbed,

or "3" highly disturbed. Any hourly value to which the numeral 3 is

attached is treated as disturbed, and the annual Potsdam publication

contains tables giving the annual and diurnal variations in the number

of such disturbed hours for D, H and V. According to this point of

view, the extent to which the hourly value departs from the mean for

that hour is immaterial to the results. It is the greater or less

sinuosity and irregularity of the curve that counts. Tables XXXIII.

and XXXIV. give an abstract of the mean Potsdam results from 1892 to

1901. The data are percentages: in Table XXXIII. of the mean monthly

total, in Table XXXIV. of the total for the day. So far as the annual

variation is concerned, the results in Table XXXIII. are fairly

similar to those in Table XXX. for Parc St Maur. There are pronounced

maxima near the equinoxes, especially the spring equinox. The diurnal

variations, however, in Tables XXXII. and XXXIV. are dissimilar. Thus

in the case of H the largest disturbance numbers at Parc St Maur

occurred between 6 a.m. and 6 p.m., whereas in Table XXXIV. they occur

between 4 p.m. and midnight. Considering the comparative proximity of

Parc St Maur and Potsdam, one must conclude that the apparent

differences between the results for these two stations are due almost

entirely to the difference in the definition of disturbance.

TABLE XXXIII.--Annual Variation of Potsdam Disturbances.

+---------+-----+-----+-----+-------+-----+------+------+-----+------+-----+------+-----+

| Element.| Jan.| Feb.| Mar.| April.| May.| June.| July.| Aug.| Sept.| Oct.| Nov.| Dec.|

+---------+-----+-----+-----+-------+-----+------+------+-----+------+-----+------+-----+

| D | 129 | 170 | 149 | 90 | 86 | 57 | 62 | 64 | 59 | 118 | 94 | 82 |

| H | 109 | 133 | 131 | 102 | 109 | 82 | 94 | 91 | 89 | 101 | 75 | 84 |

| V | 106 | 171 | 170 | 108 | 121 | 56 | 64 | 74 | 93 | 87 | 78 | 70 |

+---------+-----+-----+-----+-------+-----+------+------+-----+------+-----+------+-----+

| Mean | 115 | 158 | 150 | 100 | 105 | 65 | 73 | 76 | 94 | 102 | 82 | 79 |

+---------+-----+-----+-----+-------+-----+------+------+-----+------+-----+------+-----+

TABLE XXXIV.--Diurnal Variation of Potsdam Disturbances.

+--------+------+------+-----+---------+------+------+------+-------+

| Hours. | 1-3. | 4-6. | 7-9.| 10-noon.| 1-3. | 4-6. | 7-9. | 10-12.|

+--------+------+------+-----+---------+------+------+------+-------+

| D | 14.9 | 11.1 | 8.0 | 5.2 | 5.7 | 13.1 | 22.5 | 19.5 |

| H | 10.5 | 8.4 | 8.0 | 8.5 | 11.3 | 17.6 | 19.2 | 16.5 |

| V | 13.5 | 9.7 | 5.7 | 4.7 | 8.5 | 17.2 | 21.5 | 19.2 |

+--------+------+------+-----+---------+------+------+------+-------+

| Mean | 13.0 | 9.7 | 7.2 | 6.1 | 8.5 | 16.0 | 21.1 | 18.4 |

+--------+------+------+-----+---------+------+------+------+-------+

TABLE XXXV.--Disturbed Day less ordinary Day Inequality (Unit 1´, + to West).

+------+------+------+------+------+------+------+------+------+------+------+------+------+

| Hour.| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

+------+------+------+------+------+------+------+------+------+------+------+------+------+

| a.m. | -3.4 | -2.6 | -2.0 | -0.3 | +1.6 | +1.9 | +2.3 | +2.0 | +2.1 | +2.0 | +1.6 | +1.8 |

| p.m. | +1.8 | +2.2 | +2.1 | +1.7 | +1.4 | 0.0 | -1.3 | -2.8 | -3.5 | -2.6 | -3.5 | -2.4 |

+------+------+------+------+------+------+------+------+------+------+------+------+------+

One difficulty in the Potsdam procedure is the maintenance of a

uniform standard. Unless very frequent reference is made to the curves

of some standard year there must be a tendency to enter under "3" in

quiet years a number of hours which would be entered under "2" in a

highly disturbed year. Still, such a source of uncertainty is unlikely

to have much influence on the diurnal, or even on the annual,

variation.

§ 33. A third method of investigating a diurnal period in disturbances

is to form a diurnal inequality from disturbed days alone, and compare

it with the corresponding inequalities from ordinary or from quiet

days. Table XXXV. gives some declination data for Kew, the quantity

tabulated being the algebraic excess of the disturbed day hourly value

over that for the ordinary day in the mean diurnal inequality for the

year, as based on the 11 years 1890 to 1900.

[Illustration: FIG. 11.]

The disturbed day inequality was corrected for non-cyclic change in

the usual way. Fig. 11 shows the results of Table XXXV. graphically.

The irregularities are presumably due to the limited number, 209, of

disturbed days employed; to get a smooth curve would require probably

a considerably longer period of years. The differences between

disturbed and ordinary days at Kew are of the same general character

as those between ordinary and quiet days in Table XXIX.; they are,

however, very much larger, the range in Table XXXV. being fully 5½

times that in Table XXIX. If quiet days had replaced ordinary days in

Table XXXV., the algebraic excess of the disturbed day would have

varied from +2´.7 at 2 p.m. to -4´.1 at 11 p.m., or a range of 6´.8.

§ 34. When the mean diurnal inequality in declination for the year at

Kew is analysed into Fourier waves, the chief difference, it will be

remembered, between ordinary and quiet days was that the amplitude of

the 24-hour term was enhanced in the ordinary days, whilst its phase

angle indicated an earlier occurrence of the maximum. Similarly, the

chief difference between the Fourier waves for the disturbed and

ordinary day inequalities at Kew is the increase in the amplitude of

the 24-hour term in the former by over 70%, and the earlier occurrence

of its maximum by about 1 hour 50 minutes. It is clear from these

results for Kew, and it is also a necessary inference from the

differences obtained by Sabine's method between east and west or + and

- disturbances, that there is present during disturbances some

influence which affects the diurnal inequality in a regular systematic

way, tending to make the value of the element higher during some hours

and lower during others than it is on days relatively free from

disturbance. At Kew the consequence is a notable increase in the range

of the regular diurnal inequality on disturbed days; but whether this

is the general rule or merely a local peculiarity is a subject for

further research.

Encyclopaedia Britannica, 11th Edition, "McKinley, William" to "Magnetism, Terrestrial" Volume 17, Slice 3 · 第 44 / 45 章