Chapter LXIII: repeats the promise of freedom to the English church (10)

+---------------------------------------------------------------------------------------------+-------------------------------------+
| Declination. | Inclination. |
+-----------+---------+-----------+-----------+-----------------------+-----------+-----------+---------+---------+---------+-------+
| | Liznar, | Potsdam, | Parc St | Kew (1890-1900). | Batavia, | | Liznar &| | Parc St.| |
| | N. Hemi-| 1891-1906.| Maur, +-------+-------+-------+ 1883-1893.| Mauritius.| Hann's | Potsdam.| Maur. | Kew. |
| | 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. |
| +-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
| _Declination._ | | | | | | | | | | | | | |
| 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 |
| | | | | | | | | | | | | | |
| _Horizontal force._| | | | | | | | | | | | | |
| 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 |
| | | | | | | | | | | | | | |
| _Vertical force._ | | | | | | | | | | | | | |
| 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.

+----------+-------------------------+-----------------------------------+---------------------------------+
| | Pavlovsk. | Ekatarinburg. | Kew. |
+----------+-------+-------+---------+-------+-------+---------+---------+-------+-------+---------+-------+
| | 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. +---------------------------+---------------------------+---------------------------+
| | | Daily. | Monthly. | Yearly. |
+---------+-------+-------+-------+-------+---------+---------+-------+---------+---------+-------+---------+---------+
| | 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. |
|---------+----------------+------------+----------------+------------------+
| | Declination | | Declination | Horizontal Force |
| | (unit 1´). | | (unit 1´). | (unit 1[gamma]). |
| 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.

+---------------------------------+----------------------+----------------------+----------------------+----------------------+
| | Declination | Inclination | Horizontal Force | Vertical Force |
| | (unit 1´). | (unit 1´). | (unit 1[gamma]). | (unit 1[gamma]). |
+---------------------------------+------+------+--------+------+------+--------+------+------+--------+------+------+--------+
| 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 / | | | | | | | | | | | | |
+---------------------------------+------+------+--------+------+------+--------+------+------+--------+------+------+--------+
| _Mean from individual months:--_| | | | | | | | | | | | |
| Pavlovsk, 1890-1900 all | 6.81 |.0446 | .66 | 1.44 |.0151 | 1.05 | 22.8 | .243 | 1.07 | 9.7 | .287 | 2.97 |
| " " " quiet | 6.52 |.0442 | .68 | .. | .. | .. | 22.2 | .208 | 0.94 | 7.0 | .044 | 0.63 |
| Ekatarinburg, 1890-1900 all | 6.18 |.0355 | .58 | 1.12 |.0120 | 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.

+---------+-------------+-------------+-------------+-------------+
| | Year. | Winter. | Equinox. | Summer. |
| +------+------+------+------+------+------+------+------+
| |S max.|S min.|S max.|S min.|S max.|S min.|S max.|S min.|
+---------+------+------+------+------+------+------+------+------+
| | ´ | ´ | ´ | ´ | ´ | ´ | ´ | ´ |
| 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:--

+---------------+--------------+--------------+------------+-------------+
| | Westerly | Inclination. | Horizontal | Vertical |
| | Declination. | | Force. | Force. |
+---------------+--------------+--------------+------------+-------------+
| St Petersburg | +0.24 | -0.23 | +3.2[gamma]| -0.8[gamma] |
| Greenwich | +0.08 | | +3.2[gamma]| -0.9[gamma] |
+---------------+--------------+--------------+------------+-------------+

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. |
+------------------------------+--------+--------+--------------+--------------+
| | ´ | ´ | | |
| Mean annual change | -5.79 | -2.38 | +25.9[gamma] | -22.6[gamma] |
| Mean daily change, all days | -0.016 | -0.007 | +0.07[gamma] | -0.06[gamma] |
| Mean daily change, quiet days| +0.044 | -0.245 | +3.34[gamma] | -0.84[gamma] |
+------------------------------+------------------+-------------+--------------+

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:--

+-----------+--------------+-------------+--------------+
| | Greenwich | Falmouth | Kolaba |
| Element. | (1890-1895). | (1898-1902).| (1894-1901). |
+-----------+--------------+-------------+--------------+
| | ´ | ´ | ´ |
| D | + 0.03 | + 0.05 | + 0.07 |
| H | + 4.3[gamma] | + 3.0[gamma]| + 3.9[gamma] |
+-----------+--------------+-------------+--------------+

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.

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).