Chapter LXIII: repeats the promise of freedom to the English church (12)

+----------+-----------+-----------+-----------+-----------+-----------+-----------+
| Lunar | Kew. | Toronto. | Batavia. | St Helena.| Cape. | Hobart. |
| Hour. | 1858-1862.| 1843-1848.| 1883-1899.| 1843-1847.| 1842-1846.| 1841-1848.|
+----------+-----------+-----------+-----------+-----------+-----------+-----------+
| 0 | +103 | +315 | -70 | - 43 | -148 | - 98 |
| 1 | +160 | +275 | -63 | - 5 | -107 | -138 |
| 2 | +140 | +158 | -39 | + 37 | - 35 | -142 |
| 3 | + 33 | + 2 | - 8 | + 70 | + 43 | -107 |
| 4 | + 10 | -153 | +38 | + 85 | +108 | - 45 |
| 5 | - 67 | -265 | +63 | + 77 | +140 | + 27 |
| 6 | -150 | -302 | +87 | + 48 | +132 | + 88 |
| 7 | -188 | -255 | +77 | + 5 | + 82 | +122 |
| 8 | -160 | -137 | +40 | - 43 | + 5 | +120 |
| 9 | - 78 | + 7 | - 4 | - 82 | - 78 | + 82 |
| 10 | + 2 | +178 | -45 | -102 | -143 | + 17 |
| 11 | + 92 | +288 | -80 | - 98 | -177 | - 57 |
| 12 | +160 | +323 | -87 | - 73 | -165 | -120 |
| 13 | +188 | +272 | -68 | - 32 | -112 | -152 |
| 14 | +158 | +148 | -43 | + 13 | - 30 | -147 |
| 15 | + 90 | - 17 | - 8 | + 52 | + 58 | -105 |
| 16 | + 10 | -180 | +30 | + 73 | +132 | - 35 |
| 17 | - 85 | -297 | +62 | + 73 | +172 | + 45 |
| 18 | -142 | -337 | +72 | + 52 | +168 | +112 |
| 19 | -163 | -290 | +68 | + 17 | +122 | +152 |
| 20 | -147 | -170 | +52 | - 25 | + 45 | +152 |
| 21 | -123 | - 7 | + 8 | - 58 | - 40 | +113 |
| 22 | - 40 | +155 | -28 | - 73 | -112 | + 47 |
| 23 | + 27 | +265 | -56 | - 68 | -153 | - 30 |
+----------+-----------+-----------+-----------+-----------+-----------+-----------+
| Mean De-\| | | | | | |
| parture /| 105 | 200 | 50 | 54 | 104 | 93 |
+----------+-----------+-----------+-----------+-----------+-----------+-----------+
| Range | 376 | 660 | 174 | 187 | 349 | 304 |
+----------+-----------+-----------+-----------+-----------+-----------+-----------+

Lunar action has been specially studied in connexion with observations
from India and Java. Broun[56] at Trivandrum and C. Chambers[57] at
Kolaba investigated lunar action from a variety of aspects. At Batavia
van der Stok[58] and more recently S. Figee[59] have carried out
investigations involving an enormous amount of computation. Table
XLIII. gives a summary of Figee's results for the mean lunar diurnal
inequality at Batavia, for the two half-yearly periods April to
September (Winter or W.), and October to March (S.). The + sign
denotes movement to the west in the case of declination, but numerical
increase in the case of the other elements. In the case of H and T
(total force) the results for the two seasons present comparatively
small differences, but in the case of D, I and V the amplitude and
phase both differ widely. Consequently a mean lunar diurnal variation
derived from all the months of the year gives at Batavia, and
presumably at other tropical stations, an inadequate idea of the
importance of the lunar influence. In January Figee finds for the
range of the lunar diurnal inequality 0´.62 in D, 3.1[gamma] in H and
3.5[gamma] in V, whereas the corresponding ranges in June are only
0´.13, 1.1[gamma] and 2.2[gamma] respectively. The difference between
summer and winter is essentially due to solar action, thus the lunar
influence on terrestrial magnetism is clearly a somewhat complex
phenomenon. From a study of Trivandrum data, Broun concluded that the
action of the moon is largely dependent on the solar hour at the time,
being on the average about twice as great for a day hour as for a
night hour. Figee's investigations at Batavia point to a similar
conclusion. Following a method suggested by Van der Stok, Figee
arrives at a numerical estimate of the "lunar activity" for each hour
of the solar day, expressed in terms of that at noon taken as 100. In
summer, for instance, in the case of D he finds the "activity" varying
from 114 at 10 a.m. to only 8 at 9 p.m.; the corresponding extremes in
the case of H are 139 at 10 a.m. and 54 at 6 a.m.

TABLE XLIII.--Lunar Diurnal Inequality at Batavia in Winter and Summer.

+----------+---------------+----------------+----------------+----------------+----------------+
| | Declination | Inclination, S.| H. (unit | V. (unit | T. (unit |
| | (unit 0´.001).| (unit 0´.001). | 0.01[gamma]). | 0.01[gamma]). | 0.01[gamma]). |
+----------+-------+-------+--------+-------+--------+-------+--------+-------+--------+-------+
| Lunar | | | | | | | | | | |
| Hour. | W. | S. | W. | S. | W. | S. | W. | S. | W. | S. |
+----------+-------+-------+--------+-------+--------+-------+--------+-------+--------+-------+
| 0 | +30 | -170 | - 1 | +25 | -15 | - 56 | - 9 | + 4 | - 17 | -47 |
| 1 | +21 | -147 | -23 | +49 | -40 | - 87 | -54 | +20 | - 61 | -67 |
| 2 | + 5 | - 83 | -49 | +69 | -25 | -107 | -82 | +37 | - 62 | -76 |
| 3 | - 5 | - 12 | -51 | +47 | -21 | - 76 | -83 | +24 | - 59 | -55 |
| 4 | + 1 | + 76 | -37 | +43 | -13 | - 59 | -58 | +18 | - 39 | -38 |
| 5 | - 8 | +134 | -23 | +12 | +10 | - 9 | -27 | +11 | - 4 | - 3 |
| 6 | - 7 | +181 | - 2 | -21 | +21 | + 43 | + 9 | - 6 | + 23 | +35 |
| 7 | -10 | +164 | +30 | -12 | +23 | + 45 | +55 | + 8 | + 47 | +43 |
| 8 | - 7 | + 86 | +36 | -21 | +38 | + 52 | +71 | - 1 | + 68 | +45 |
| 9 | - 8 | 0 | +28 | -23 | +46 | + 30 | +64 | -16 | + 71 | +19 |
| 10 | - 5 | - 85 | +34 | -20 | +13 | + 13 | +54 | -21 | + 38 | + 1 |
| 11 | -15 | -144 | +27 | -11 | -12 | - 6 | +31 | -19 | + 5 | -15 |
| 12 | - 9 | -164 | +19 | - 5 | -47 | - 23 | 0 | -19 | - 41 | -29 |
| 13 | + 1 | -136 | - 3 | +17 | -59 | - 46 | -36 | - 2 | - 69 | -41 |
| 14 | - 7 | - 79 | -13 | +27 | -66 | - 44 | -55 | +14 | - 84 | -32 |
| 15 | - 8 | - 8 | -32 | +25 | -53 | - 37 | -74 | +14 | - 82 | -26 |
| 16 | -12 | + 72 | -37 | +25 | -34 | - 17 | -70 | +26 | - 64 | - 2 |
| 17 | -13 | +137 | -33 | + 4 | - 1 | + 28 | -47 | +21 | - 24 | +35 |
| 18 | -21 | +165 | - 2 | -10 | +20 | + 47 | + 8 | +12 | + 21 | +47 |
| 19 | -12 | +147 | +21 | -42 | +44 | + 81 | +53 | -14 | + 64 | +64 |
| 20 | +10 | + 95 | +21 | -62 | +75 | +107 | +71 | -28 | +100 | +80 |
| 21 | +13 | + 4 | +26 | -70 | +65 | + 98 | +72 | -44 | + 92 | +65 |
| 22 | +25 | - 82 | +35 | -41 | +35 | + 35 | +68 | -38 | + 64 | +12 |
| 23 | +36 | -147 | +34 | - 4 | - 7 | - 14 | +44 | -13 | + 15 | -19 |
+----------+-------+-------+--------+-------+--------+-------+--------+-------+--------+-------+
| Mean De-\| | | | | | | | | | |
| parture /| 12 | 150 | 26 | 29 | 33 | 48 | 50 | 18 | 51 | 37 |
+----------+-------+-------+--------+-------+--------+-------+--------+-------+--------+-------+
| Range | 57 | 351 | 87 | 139 | 141 | 214 | 155 | 81 | 184 | 156 |
+----------+-------+-------+--------+-------+--------+-------+--------+-------+--------+-------+

The question whether lunar influence increases with sun-spot frequency
is obviously of considerable theoretical interest. Balfour Stewart in
the 9th edition of this encyclopaedia gave some data indicating an
appreciably enhanced lunar influence at Trivandrum during years of
sun-spot maximum, but he hesitated to accept the result as finally
proved. Figee recently investigated this point at Batavia, but with
inconclusive results. Attempts have also been made to ascertain how
lunar influence depends on the moon's declination and phase, and on
her distance from the earth. The difficulty in these investigations is
that we are dealing with a small effect, and a very long series of
data would be required satisfactorily to eliminate other periodic
influences.

Planetary Influence.

§ 41. From an analysis of seventeen years data at St Petersburg and
Pavlovsk, Leyst[60] concluded that all the principal planets sensibly
influence the earth's magnetism. According to his figures, all the
planets except Mercury--whose influence he found opposite to that of
the others--when nearest the earth tended to deflect the declination
magnet at St Petersburg to the west, and also increased the range of
the diurnal inequality of declination, the latter effect being the
more conspicuous. Schuster,[61] who has considered the evidence
advanced by Leyst from the mathematical standpoint, considers it to be
inconclusive.

Magnetic Surveys.

§ 42. The best way of carrying out a magnetic survey depends on where
it has to be made and on the object in view. The object that probably
still comes first in importance is a knowledge of the declination, of
sufficient accuracy for navigation in all navigable waters. One might
thus infer that magnetic surveys consist mainly of observations at
sea. This cannot however be said to be true of the past, whatever it
may be of the future, and this for several reasons. Observations at
sea entail the use of a ship, specially constructed so as to be free
from disturbing influence, and so are inherently costly; they are
also apt to be of inferior accuracy. It might be possible in quiet
weather, in a large vessel free from vibration, to observe with
instruments of the highest precision such as a unifilar magnetometer,
but in the ordinary surveying ship apparatus of less sensitiveness has
to be employed. The declination is usually determined with some form
of compass. The other elements most usually found directly at sea are
the inclination and the total force, the instrument employed being a
special form of inclinometer, such as the Fox circle, which was
largely used by Ross in the Antarctic, or in recent years the
Lloyd-Creak. This latter instrument differs from the ordinary
dip-circle fitted for total force observations after H. Lloyd's method
mainly in that the needles rest in pivots instead of on agate edges.
To overcome friction a projecting pin on the framework is scratched
with a roughened ivory plate.

The most notable recent example of observations at sea is afforded by
the cruises of the surveying ships "Galilee" and "Carnegie" under the
auspices of the Carnegie Institution of Washington, which includes in
its magnetic programme a general survey. To see where the ordinary
land survey assists navigation, let us take the case of a country with
a long seaboard. If observations were taken every few miles along the
coast results might be obtained adequate for the ordinary wants of
coasting steamers, but it would be difficult to infer what the
declination would be 50 or even 20 miles off shore at any particular
place. If, however, the land area itself is carefully surveyed, one
knows the trend of the lines of equal declination, and can usually
extend them with considerable accuracy many miles out to sea. One also
can tell what places if any on the coast suffer from local
disturbances, and thus decide on the necessity of special
observations. This is by no means the only immediately useful purpose
which is or may be served by magnetic surveys on land. In Scandinavia
use has been made of magnetic observations in prospecting for iron
ore. There are also various geological and geodetic problems to whose
solution magnetic surveys may afford valuable guidance. Among the most
important recent surveys may be mentioned those of the British Isles
by A. Rücker and T. E. Thorpe,[62] of France and Algeria by
Moureaux,[63] of Italy by Chistoni and Palazzo,[64] of the Netherlands
by Van Ryckevorsel,[65] of South Sweden by Carlheim Gyllenskiöld,[66]
of Austria-Hungary by Liznar,[67] of Japan by Tanakadate,[68] of the
East Indies by Van Bemmelen, and South Africa by J. C. Beattie. A
survey of the United States has been proceeding for a good many years,
and many results have appeared in the publications of the U.S. Coast
and Geodetic Survey, especially Bauer's _Magnetic Tables and Magnetic
Charts_, 1908. Additions to our knowledge may also be expected from
surveys of India, Egypt and New Zealand.

For the satisfactory execution of a land survey, the observers must
have absolute instruments such as the unifilar magnetometer and dip
circle, suitable for the accurate determination of the magnetic
elements, and they must be able to fix the exact positions of the
spots where observations are taken. If, as usual, the survey occupies
several years, what is wanted is the value of the elements not at the
actual time of observation, but at some fixed epoch, possibly some
years earlier or later. At a magnetic observatory, with standardized
records, the difference between the values of a magnetic element at
any two specified instants can be derived from the magnetic curves.
But at an ordinary survey station, at a distance from an observatory,
the information is not immediately available. Ordinarily the reduction
to a fixed epoch is done in at least two stages, a correction being
applied for secular change, and a second for the departure from the
mean value for the day due to the regular diurnal inequality and to
disturbance.

The reduction to a fixed epoch is at once more easy and more accurate
if the area surveyed contains, or has close to its borders, a well
distributed series of magnetic observatories, whose records are
comparable and trustworthy. Throughout an area of the size of France
or Germany, the secular change between any two specified dates can
ordinarily be expressed with sufficient accuracy by a formula of the
type

[delta] = [delta]0 + a(l - l0) + b([lambda] - [lambda]0) (i),

where [delta] denotes secular change, l latitude and [lambda]
longitude, the letters with suffix _0_ relating to some convenient
central position. The constants [delta]0, a, b are to be determined
from the observed secular changes at the fixed observatories whose
geographical co-ordinates are accurately known. Unfortunately, as a
rule, fixed observatories are few in number and not well distributed
for survey purposes; thus the secular change over part at least of the
area has usually to be found by repeating the observations after some
years at several of the field stations. The success attending this
depends on the exactitude with which the sites can be recovered, on
the accuracy of the observations, and on the success with which
allowance is made for diurnal changes, regular and irregular. It is
thus desirable that the observations at repeat stations should be
taken at hours when the regular diurnal changes are slow, and that
they should not be accepted unless taken on days that prove to be
magnetically quiet. Unless the secular change is exceptionally rapid,
it will usually be most convenient in practice to calculate it from or
to the middle of the month, and then to allow for the difference
between the mean value for the month and the value at the actual hour
of observation. There is here a difficulty, inasmuch as the latter
part of the correction depends on the diurnal inequality, and so on
the local time of the station. No altogether satisfactory method of
surmounting this difficulty has yet been proposed. Rücker and Thorpe
in their British survey assumed that the divergence from the mean
value at any hour at any station might be regarded as made up of a
regular diurnal inequality, identical with that at Kew when both were
referred to _local_ time, and of a disturbance element identical with
that existing at the same absolute time at Kew. Suppose, for instance,
that at hour h G.M.T. the departure at Kew from the mean value for the
month is d, then the corresponding departure from the mean at a
station [lambda] degrees west of Kew is d - e, where e is the increase
in the element at Kew due to the regular diurnal inequality between
hour h - [lambda]/15 and hour h. This procedure is simple, but is
exposed to various criticisms. If we define a diurnal inequality as
the result obtained by combining hourly readings from all the days of
a month, we can assign a definite meaning to the diurnal inequality
for a particular month of a particular year, and after the curves have
been measured we can give exact numerical figures answering to this
definition. But the diurnal inequality thus obtained differs, as has
been pointed out, from that derived from a limited number of the
quietest days of the month, not merely in amplitude but in phase, and
the view that the diurnal changes on any individual day can be
regarded as made up of a regular diurnal inequality of definite
character and of a disturbance element is an hypothesis which is
likely at times to be considerably wide of the mark. The extent of the
error involved in assuming the regular diurnal inequality the same in
the north of Scotland, or the west of Ireland, as in the south-east of
England remains to be ascertained. As to the disturbance element, even
if the disturbing force were of given magnitude and direction all over
the British Isles--which we now know is often very far from the
case--its effects would necessarily vary very sensibly owing to the
considerable variation in the direction and intensity of the local
undisturbed force. If observations were confined to hours at which the
regular diurnal changes are slow, and only those taken on days of
little or no disturbance were utilized, corrections combining the
effects of regular and irregular diurnal changes could be derived from
the records of fixed observations, supposed suitably situated,
combined in formulae of the same type as (i).

§ 43. The field results having been reduced to a fixed epoch, it
remains to combine them in ways likely to be useful. In most cases the
results are embodied in charts, usually of at least two kinds, one set
showing only general features, the other the chief local
peculiarities. Charts of the first kind resemble the world charts
(figs. 1 to 4) in being free from sharp twistings and convolutions. In
these the declination for instance at a fixed geographical position on
a particular isogonal is to be regarded as really a mean from a
considerable surrounding area.

Various ways have been utilized for arriving at these _terrestrial
isomagnetics_--as Rücker and Thorpe call them--of which an elaborate
discussion has been made by E. Mathias.[69] From a theoretical
standpoint the simplest method is perhaps that employed by Liznar for
Austria-Hungary. Let l and [lambda] represent latitude and longitude
relative to a certain central station in the area. Then assume that
throughout the area the value E of any particular magnetic element is
given by a formula

E = E0 + al + b[lambda] + cl² + d[lambda]² + el[lambda],

where E0, a, b, c, d, e are absolute constants to be determined from
the observations. When determining the constants, we write for E in
the equation the observed value of the element (corrected for secular
change, &c.) at each station, and for l and [lambda] the latitude and
longitude of the station relative to the central station. Thus each
station contributes an equation to assist in determining the six
constants. They can thus be found by least squares or some simpler
method. In Liznar's case there were 195 stations, so that the labour
of applying least squares would be considerable. This is one objection
to the method. A second is that it may allow undesirably large weight
to a few highly disturbed stations. In the case of the British Isles,
Rücker and Thorpe employed a different method. The area was split up
into _districts_. For each district a mean was formed of the observed
values of each element, and the mean was assigned to an imaginary
central station, whose geographical co-ordinates represented the mean
of the geographical co-ordinates of the actual stations. Want of
uniformity in the distribution of the stations may be allowed for by
weighting the results. Supposing E0 the value of the element found for
the central station of a district, it was assumed that the value E at
any actual station whose latitude and longitude exceeded those of the
central station by l and [lambda] was given by E = E0 + al +
b[lambda], with a and b constants throughout the district. Having
found E0, a and b, Rücker and Thorpe calculated values of the element
for points defined by whole degrees of longitude (from Greenwich) and
half degrees of latitude. Near the common border of two districts
there would be two calculated values, of which the arithmetic mean was
accepted.

The next step was to determine by interpolation where isogonals--or
other isomagnetic lines--cut successive lines of latitude. The curves
formed by joining these successive points of intersection were called
_district_ lines or curves. Rücker and Thorpe's next step was to
obtain formulae by trial, giving smooth curves of continuous
curvature--terrestrial isomagnetics--approximating as closely as
possible to the district lines. The curves thus obtained had somewhat
complicated formulae. For instance, the isogonals south of 54°.5
latitude were given for the epoch Jan. 1, 1891 by

D = 18° 37´ + 18´.5(l - 49.5) - 3´.5 cos {45°(l - 49.5)}
+ {26´.3 + 1´.5(l - 49.5)} ([lambda] - 4)
+ 0´.01([lambda] - 4)² (l - 54.5)²,

where D denotes the westerly declination. Supposing, what is at least
approximately true, that the secular change in Great Britain since
1891 has been uniform south of lat. 54°.5, corresponding formulae for
the epochs Jan. 1, 1901, and Jan. 1, 1906, could be obtained by
substituting for 18° 37´ the values 17° 44´ and 17° 24´ respectively.
In their very laborious and important memoir E. Mathias and B.
Baillaud[69] have applied to Rücker and Thorpe's observations a method
which is a combination of Rücker and Thorpe's and of Liznar's. Taking
Rücker and Thorpe's nine districts, and the magnetic data found for
the nine imaginary central stations, they employed these to determine
the six constants of Liznar's formula. This is an immense
simplification in arithmetic. The declination formula thus obtained
for the epoch Jan. 1, 1891, was

D = 20° 45´.89 + .53474[lambda] + .34716l + .000021[lambda]²
+ .000343l[lambda] - .000239l²,

where l + (53° 30´.5) represents the latitude, and ([lambda] + 5°
35´.2) the west longitude of the station. From this and the
corresponding formulae for the other elements, values were calculated
for each of Rücker and Thorpe's 882 stations, and these were compared
with the observed values. A complete record is given of the
differences between the observed and calculated values, and of the
corresponding differences obtained by Rücker and Thorpe from their own
formulae. The mean numerical (calculated ~ observed) differences from
the two different methods are almost exactly the same--being
approximately 10´ for declination, 5´½ for inclination, and 70[gamma]
for horizontal force. The applications by Mathias[69] of his method to
the survey data of France obtained by Moureaux, and those of the
Netherlands obtained by van Rïjckevorsel, appear equally successful.
The method dispenses entirely with district curves, and the parabolic
formulae are perfectly straightforward both to calculate and to apply;
they thus appear to possess marked advantages. Whether the method
could be applied equally satisfactorily to an area of the size of
India or the United States actual trial alone would show.

Local Disturbances.

§ 44. Rücker and Thorpe regarded their terrestrial isomagnetics and
the corresponding formulae as representing the normal field that would
exist in the absence of disturbances peculiar to the neighbourhood.
Subtracting the forces derived from the formulae from those observed,
we obtain forces which may be ascribed to regional disturbance.

When the vertical disturbing force is downwards, or the observed
vertical component larger than the calculated, Rücker and Thorpe
regard it as positive, and the loci where the largest positive values
occur they termed _ridge lines_. The corresponding loci where the
largest negative values occur were called _valley lines_. In the
British Isles Rücker and Thorpe found that almost without exception,
in the neighbourhood of a ridge line, the horizontal component of the
disturbing force pointed towards it, throughout a considerable area on
both sides. The phenomena are similar to what would occur if ridge
lines indicated the position of the summits of underground masses of
magnetic material, magnetized so as to attract the north-seeking pole
of a magnet. Rücker and Thorpe were inclined to believe in the real
existence of these subterranean magnetic mountains, and inferred that
they must be of considerable extent, as theory and observation alike
indicate that thin basaltic sheets or dykes, or limited masses of trap
rock, produce no measurable magnetic effect except in their immediate
vicinity. In support of their conclusions, Rücker and Thorpe dwell on
the fact that in the United Kingdom large masses of basalt such as
occur in Skye, Mull, Antrim, North Wales or the Scottish coalfield,
are according to their survey invariably centres of attraction for the
north-seeking pole of a magnet. Various cases of repulsion have,
however, been described by other observers in the northern hemisphere.

§ 45. Rücker and Thorpe did not make a very minute examination of
disturbed areas, so that purely local disturbances larger than any
noticed by them may exist in the United Kingdom. But any that exist
are unlikely to rival some that have been observed elsewhere, notably
those in the province of Kursk in Russia described by Moureaux[70] and
by E. Leyst.[71] In Kursk Leyst observed declinations varying from 0°
to 360°, inclinations varying from 39°.1 to 90°; he obtained values of
the horizontal force varying from 0 to 0.856 C.G.S., and values of the
vertical force varying from 0.371 to 1.836. Another highly disturbed
Russian district Krivoi Rog (48° N. lat. 33° E. long.) was
elaborately surveyed by Paul Passalsky.[72] The extreme values
observed by him differed, the declination by 282° 40´, the inclination
by 41° 53´, horizontal force by 0.658, and vertical force by 1.358. At
one spot a difference of 116°½ was observed between the declinations
at two positions only 42 metres apart. In cases such as the last
mentioned, the source of disturbance comes presumably very near the
surface. It is improbable that any such enormously rapid changes of
declination can be experienced anywhere at the surface of a deep
ocean. But in shallow water disturbances of a not very inferior order
of magnitude have been met with. Possibly the most outstanding case
known is that of an area, about 3 m. long by 1¼ m. at its widest, near
Port Walcott, off the N.W. Australian coast. The results of a minute
survey made here by H.M.S. "Penguin" have been discussed by Captain E.
W. Creak.[73] Within the narrow area specified, declination varied
from 26° W. to 56° E., and inclination from 50° to nearly 80°, the
observations being taken some 80 ft. above sea bottom. Another
noteworthy case, though hardly comparable with the above, is that of
East Loch Roag at Lewis in the Hebrides. A survey by H.M.S. "Research"
in water about 100 ft. deep--discussed by Admiral A. M.
Field[74]--showed a range of 11° in declination. The largest observed
disturbances in horizontal and vertical force were of the order 0.02
and 0.05 C.G.S. respectively. An interesting feature in this case was
that vertical force was reduced, there being a well-marked valley
line.

In some instances regional magnetic disturbances have been found to be
associated with geodetic anomalies. This is true of an elongated area
including Moscow, where observations were taken by Fritsche.[75]
Again, Eschenhagen[76] detected magnetic anomalies in an area
including the Harz Mountains in Germany, where deflections of the
plumb line from the normal had been observed. He found a magnetic
ridge line running approximately parallel to the line of no deflection
of the plumb line.

§ 46. A question of interest, about which however not very much is
known, is the effect of local disturbance on secular change and on the
diurnal inequality. The determination of secular change in a highly
disturbed locality is difficult, because an unintentional slight
change in the spot where the observations are made may wholly falsify
the conclusions drawn. When the disturbed area is very limited in
extent, the magnetic field may reasonably be regarded as composed of
the normal field that would have existed in the absence of local
disturbance, plus a disturbance field arising from magnetic material
which approaches nearly if not quite to the surface. Even if no
sensible change takes place in the disturbance field, one would hardly
expect the secular change to be wholly normal. The changes in the
rectangular components of the force may possibly be the same as at a
neighbouring undisturbed station, but this will not give the same
change in declination and inclination. In the case of the diurnal
inequality, the presumption is that at least the declination and
inclination changes will be influenced by local disturbance. If, for
example, we suppose the diurnal inequality to be due to the direct
influence of electric currents in the upper atmosphere, the
declination change will represent the action of the component of a
force of given magnitude which is perpendicular to the position of the
compass needle. But when local disturbance exists, the direction of
the needle and the intensity of the controlling field are both altered
by the local disturbance, so it would appear natural for the
declination changes to be influenced also. This conclusion seems borne
out by observations made by Passalsky[72] at Krivoi Rog, which showed
diurnal inequalities differing notably from those experienced at the
same time at Odessa, the nearest magnetic observatory. One station
where the horizontal force was abnormally low gave a diurnal range of
declination four times that at Odessa; on the other hand, the range of
the horizontal force was apparently reduced. It would be unsafe to
draw general conclusions from observations at two or three stations,
and much completer information is wanted, but it is obviously
desirable to avoid local disturbance when selecting a site for a
magnetic observatory, assuming one's object is to obtain data
reasonably applicable to a large area. In the case of the older
observatories this consideration seems sometimes to have been lost
sight of. At Mauritius, for instance, inside of a circle of only 56
ft. radius, having for centre the declination pillar of the absolute
magnetic hut of the Royal Alfred Observatory, T. F. Claxton[77] found
that the declination varied from 4° 56´ to 13° 45´ W., the inclination
from 50° 21´ to 58° 34´ S., and the horizontal force from 0.197 to
0.244 C.G.S. At one spot he found an alteration of 1°1/3 in the
declination when the magnet was lowered from 4 ft. above the ground to
2. Disturbances of this order could hardly escape even a rough
investigation of the site.

Gaussian Potential and Constants.

§ 47. If we assume the magnetic force on the earth's surface derivable
from a potential V, we can express V as the sum of two series of solid
spherical harmonics, one containing negative, the other positive
integral powers of the radius vector r from the earth's centre. Let
[lambda] denote east longitude from Greenwich, and let µ = cos(½[pi] -
l), where l is latitude; and also let
_ _
| (n - m)(n - m - 1) |
H_n^m = (1 - µ²)^(½m) |µ^(n-m) - ------------------ µ^(n-m-2) + ...|,
|_ 2(2n - 1) _|

where n and m denote any positive integers, m being not greater than
n. Then denoting the earth's radius by R, we have

V/R = [Sigma](R/r)^(n+1) [H_n^m (g_n^m cos m[lambda] + h_n^m sin m[lambda])]
+ [Sigma](r/R)^n [H_n^m (g_(-n)^m cos m[lambda] + h_(-n)^m sin m[lambda])],

where [Sigma] denotes summation of m from 0 to n, followed by
summation of n from 0 to [infinity]. In this equation g_n^m, &c. are
constants, those with positive suffixes being what are generally
termed _Gaussian constants_. The series with negative powers of r
answers to forces with a source internal to the earth, the series with
positive powers to forces with an external source. Gauss found that
forces of the latter class, if existent, were very small, and they are
usually left out of account. There are three Gaussian constants of the
first order, g1^0, g1¹, h1¹, five of the second order, seven of the
third, and so on. The coefficient of a Gaussian constant of the n^th
order is a spherical harmonic of the n^th degree. If R be taken as
unit length, as is not infrequent, the first order terms are given by

V1 = r^(-2) [g1^0 sin l + (g1¹ cos [lambda] + h1¹ sin [lambda]) cos l].

The earth is in reality a spheroid, and in his elaborate work on the
subject J. C. Adams[78] develops the treatment appropriate to this
case. Here we shall as usual treat it as spherical. We then have for
the components of the force at the surface

X = -R^(-1)(1 - µ²)^½ (dV/dµ) towards the astronomical north,
Y = -R^(-1)(1 - µ²)^-½ (dV/d[lambda]) " " " west,
Z = -dV/dr vertically downwards.

Supposing the Gaussian constants known, the above formulae would give
the force all over the earth's surface. To determine the Gaussian
constants we proceed of course in the reverse direction, equating the
observed values of the force components to the theoretical values
involving g_n^m, &c. If we knew the values of the component forces at
regularly distributed stations all over the earth's surface, we could
determine each Gaussian constant independently of the others. Our
knowledge however of large regions, especially in the Arctic and
Antarctic, is very scanty, and in practice recourse is had to methods
in which the constants are not determined independently. The
consequence is unfortunately that the values found for some of the
constants, even amongst the lower orders, depend very sensibly on how
large a portion of the polar regions is omitted from the calculations,
and on the number of the constants of the higher orders which are
retained.

TABLE XLIV.--Gaussian Constants of the First Order.

+------+---------+---------+---------+---------+---------+---------+---------+
| | 1829 | | | | | | |
| | Erman- | 1830 | 1845 | 1880 | 1885 | 1885 | 1885 |
| |Petersen.| Gauss. | Adams. | Adams. |Neumayer.| Schmidt.|Fritsche.|
+------+---------+---------+---------+---------+---------+---------+---------+
| g1^0 | +.32007 | +.32348 | +.32187 | +.31684 | +.31572 | +.31735 | +.31635 |
| g1^1 | +.02835 | +.03111 | +.02778 | +.02427 | +.02481 | +.02356 | +.02414 |
| h1^1 | -.06011 | -.06246 | -.05783 | -.06030 | -.06026 | -.05984 | -.05914 |
+------+---------+---------+---------+---------+---------+---------+---------+

Table XLIV. gives the values obtained for the Gaussian constants of
the first order in some of the best-known computations, as collected
by W. G. Adams.[79]

§ 48. Allowance must be made for the difference in the epochs, and for
the fact that the number of constants assumed to be worth retaining
was different in each case. Gauss, for instance, assumed 24 constants
sufficient, whilst in obtaining the results given in the table J. C.
Adams retained 48. Some idea of the uncertainty thus arising may be
derived from the fact that when Adams assumed 24 constants sufficient,
he got instead of the values in the table the following:--

g1^0 g1¹ h1¹

1842-1845 +.32173 +.02833 -.05820
1880 +.31611 +.02470 -.06071

Some of the higher constants were relatively much more affected. Thus,
on the hypotheses of 48 and of 24 constants respectively, the values
obtained for g2^0 in 1842-1845 were -.00127 and -.00057, and those
obtained for h3¹ in 1880 were +.00748 and +.00573. It must also be
remembered that these values assume that the series in positive powers
of r, with coefficients having negative suffixes, is absolutely
non-existent. If this be not assumed, then in any equation determing X
or Y, g_n^m must be replaced by g_n^m + g_(-n)^m, and in any equation
determining Z by g_n^m - n/(n + 1) g_n^m; similar remarks apply to
h_n^m and h_(-n)^m. It is thus theoretically possible to check the
truth of the assumption that the positive power series is non-existent
by comparing the values obtained for g_n^m and h_n^m from the X and Y
or from the Z equations, when g_(-n)^m and h_(-n)^m are assumed zero.
If the values so found differ, values can be found for g_(-n)^m and
h_(-n)^m which will harmonize the two sets of equations. Adams gives
the values obtained from the X, Y and the Z equations separately for
the Gaussian constants. The following are examples of the values
thence deducible for the coefficients of the positive power series:--

g_(-1)^0 g_(-1)^1 h_(-1)^1 g_(-4)^0 g_(-5)^0 g_(-6)^0

1842-1845 +.0018 -.0002 -.0014 +.0064 +.0072 +.0124
1880 -.0002 -.0012 +.0015 -.0043 -.0021 -.0013

Compared to g4^0, g5^0 and g6^0 the values here found for g_(-4)^0,
g_(-5)^0 and g_(-6)^0 are far from insignificant, and there would be
no excuse for neglecting them if the observational data were
sufficient and reliable. But two outstanding features claim attention,
first the smallness of g_(-1)^0, g_(-1)^1 and h_(-1)^1, the
coefficients least likely to be affected by observational
deficiencies, and secondly the striking dissimilarity between the
values obtained for the two epochs. The conclusion to which these and
other facts point is that observational deficiencies, even up to the
present date, are such that no certain conclusion can be drawn as to
the existence or non-existence of the positive power series. It is
also to be feared that considerable uncertainties enter into the
values of most of the Gaussian constants, at least those of the higher
orders. The introduction of the positive power series necessarily
improves the agreement between observed and calculated values of the
force, but it is more likely than not to be disadvantageous
physically, if the differences between observed values and those
calculated from the negative power series alone arise in large measure
from observational deficiencies.

TABLE XLV.--Axis and Moment of First Order Gaussian Coefficients.

+-------+-------------------+-----------+------------+--------------+
| Epoch.| Authority for | North | West | M/R³ in |
| | Constants. | Latitude. | Longitude. | G.C.S. units.|
+-------+-------------------+-----------+------------+--------------+
| | | ° ´ | ° ´ | |
| 1650 | H. Fritsche | 82 50 | 42 55 | .3260 |
| 1836 | " | 78 27 | 63 35 | .3262 |
| 1845 | J. C. Adams | 78 44 | 64 20 | .3282 |
| 1880 | " | 78 24 | 68 4 | .3234 |
| 1885 | Neumayer-Petersen | | | |
| | and Bauer | 78 3 | 67 3 | .3224 |
| 1885 | Neumayer, Schmidt | 78 34 | 68 31 | .3230 |
+-------+-------------------+-----------+------------+--------------+

§ 49. The first order Gaussian constants have a simple physical
meaning. The terms containing them represent the potential arising
from the uniform magnetization of a sphere parallel to a fixed axis,
the moment M of the spherical magnet being given by

M = R³{(g1^0)² + (g1^1)² + (h1^1)²}^½,

where R is the earth's radius. The position of the north end of the
axis of this uniform magnetization and the values of M/R^3, derived
from the more important determinations of the Gaussian constants, are
given in Table XLV. The data for 1650 are of somewhat doubtful value.
If they were as reliable as the others, one would feel greater
confidence in the reality of the apparent movement of the north end of
the axis from east to west. The table also suggests a slight
diminution in M since 1845, but it is open to doubt whether the
apparent change exceeds the probable error in the calculated values.
It should be carefully noticed that the data in the table apply only
to the first order Gaussian terms, and so only to a portion of the
earth's magnetization, and that the Gaussian constants have been
calculated on the assumption that the negative power series alone
exists. The field answering to the first order terms--or what Bauer
has called the _normal_ field--constitutes much the most important
part of the whole magnetization. Still what remains is very far from
negligible, save for rough calculations. It is in fact one of the weak
points in the Gaussian analysis that when one wishes to represent the
observed facts with high accuracy one is obliged to retain so many
terms that calculation becomes burdensome.

Earth-air Currents.

§ 50. The possible existence of a positive power series is not the
only theoretical uncertainty in the Gaussian analysis. There is the
further possibility that part of the earth's magnetic field may not
answer to a potential at all. Schmidt[80] in his calculation of
Gaussian constants regarded this as a possible contingency, and the
results he reached implied that as much as 2 or 3% of the entire field
had no potential. If the magnetic force F on the earth's surface comes
from a potential, then the line integral [int]F ds taken round any
closed circuit s should vanish. If the integral does not vanish, it
equals 4[pi]I, where I is the total electric current traversing the
area bounded by s. A + sign in the result of the integration means
that the current is downwards (i.e. from air to earth) or upwards,
according as the direction of integration round the circuit, as viewed
by an observer above ground, has been clockwise or anti-clockwise. In
applications of the formula by W. von Bezold[81] and Bauer[82] the
integral has been taken along parallels of latitude in the direction
west to east. In this case a + sign indicates a resultant upward
current over the area between the parallel of latitude traversed and
the north geographical pole. The difference between the results of
integration round two parallels of latitude gives the total vertical
current over the zone between them. Schmidt's final estimate of the
average intensity of the earth-air current, irrespective of sign, for
the epoch 1885 was 0.17 ampere per square kilometre. Bauer employing
the same observational data as Schmidt, reached somewhat similar
conclusions from the differences between integrals taken round
parallels of latitude at 5° intervals from 60° N. to 60° S. H.
Fritsche[83] treating the problem similarly, but for two epochs, 1842
and 1885, got conspicuously different results for the two epochs,
Bauer[84] has more recently repeated his calculations, and for three
epochs, 1842-1845 (Sabine's charts), 1880 (Creak's charts), and 1885
(Neumayer's charts), obtaining the mean value of the current per sq.
km. for 5° zones. Table XLVI. is based on Bauer's figures, the unit
being 0.001 ampere, and + denoting an _upwardly_ directed current.

TABLE XLVI.--Earth-air Currents, after Bauer.

+-------------+----------------------+---------------------+
| | Northern Hemisphere. | Southern Hemisphere.|
| Latitude. +--------+------+------+-------+------+------+
| | 1842-5.| 1880.| 1885.|1842-5.| 1880.| 1885.|
+-------------+--------+------+------+-------+------+------+
| 0° to 15° | - 1 | -32 | -34 | +66 | + 30 | + 36 |
| 15° " 30° | -70 | -59 | -68 | + 2 | - 62 | - 63 |
| 30° " 45° | + 3 | +14 | -22 | +26 | - 11 | - 14 |
| 45° " 60° | -31 | -21 | +78 | + 5 | +276 | +213 |
+-------------+--------+------+------+-------+------+------+

In considering the significance of the data in Table XLVI., it should
be remembered that the currents must be regarded as mean values
derived from all hours of the day, and all months of the year.
Currents which were upwards during certain hours of the day, and
downwards during others, would affect the diurnal inequality; while
currents which were upwards during certain months, and downwards
during others, would cause an annual inequality in the absolute
values. Thus, if the figures be accepted as real, we must suppose that
between 15° N. and 30° N. there are preponderatingly downward
currents, and between 0° S. and 15° S. preponderatingly upward
currents. Such currents might arise from meteorological conditions
characteristic of particular latitudes, or be due to the relative
distribution of land and sea; but, whatever their cause, any
considerable real change in their values between 1842 and 1885 seems
very improbable. The most natural cause to which to attribute the
difference between the results for different epochs in Table XLVI. is
unquestionably observational deficiencies. Bauer himself regards the
results for latitudes higher than 45° as very uncertain, but he seems
inclined to accept the reality of currents of the average intensity of
1/30 ampere per sq. km. between 45° N. and 45° S.

Currents of the size originally deduced by Schmidt, or even those of
Bauer's latest calculations, seem difficult to reconcile with the
results of atmospheric electricity (q.v.).

§ 51. There is no single parallel of latitude along the whole of which
magnetic elements are known with high precision. Thus results of
greater certainty might be hoped for from the application of the line
integral to well surveyed countries. Such applications have been made,
e.g. to Great Britain by Rücker,[85] and to Austria by Liznar,[86] but
with negative results. The question has also been considered in detail
by Tanakadate[68] in discussing the magnetic survey of Japan. He makes
the criticism that the taking of a line integral round the _boundary_
of a surveyed area amounts to utilizing the values of the magnetic
elements where least accurately known, and he thus considers it
preferable to replace the line integral by the surface integral.

4[pi]I = [int][int] (dY/dx - dX/dy) dx dy.

He applied this formula not merely to his own data for Japan, but also
to British and Austrian data of Rücker and Thorpe and of Liznar. The
values he ascribes to X and Y are those given by the formulae
calculated to fit the observations. The result reached was "a line of
no current through the middle of the country; in Japan the current is
upward on the Pacific side and downward on the Siberian side; in
Austria it is upward in the north and downward in the south; in Great
Britain upward in the east and downward in the west." The results
obtained for Great Britain differed considerably according as use was
made of Rücker and Thorpe's own district equations or of a series of
general equations of the type subsequently utilized by Mathias.
Tanakadate points out that the fact that his investigations give in
each case a line of no current passing through the middle of the
surveyed area, is calculated to throw doubt on the reality of the
supposed earth-air currents, and he recommends a suspension of
judgment.

§ 52. A question of interest, and bearing a relationship to the
Gaussian analysis, is the law of variation of the magnetic elements
with height above sea-level. If F represent the value at sea-level,
and F + [delta]F that at height h, of any component of force answering
to Gaussian constants of the n^th order, then 1 + [delta]F/F = (1 +
h/R)^(-n-2), where R is the earth's radius. Thus at heights of only a
few miles we have very approximately [delta]F/F = -(n + 2)h/R. As we
have seen, the constants of the first order are much the most
important, thus we should expect as a first approximation [delta]X/X =
[delta]Y/Y = [delta]Z/Z = -3h/R. This equation gives the same rate of
decrease in all three components, and so no change in declination or
inclination. Liznar[86a] compared this equation with the observed
results of his Austrian survey, subdividing his stations into three
groups according to altitude. He considered the agreement not
satisfactory. It must be remembered that the Gaussian analysis,
especially when only lower order terms are retained, applies only to
the earth's field freed from local disturbances. Now observations at
individual high level stations may be seriously influenced not merely
by regional disturbances common to low level stations, but by magnetic
material in the mountain itself. A method of arriving at the vertical
change in the elements, which theoretically seems less open to
criticism, has been employed by A. Tanakadate.[68] If we assume that a
potential exists, or if admitting the possibility of earth-air
currents we assume their effort negligible, we have dX/dz = dZ/dx,
dY/dz = dZ/dy. Thus from the observed rates of change of the vertical
component of force along the parallels of latitude and longitude, we
can deduce the rate of change in the vertical direction of the two
rectangular components of horizontal force, and thence the rates of
change of the horizontal force and the declination. Also we have dZ/dz
= 4[pi][rho] - (dX/dx + dY/dy), where [rho] represents the density of
free magnetism at the spot. The spot being above ground we may neglect
[rho], and thus deduce the variation in the vertical direction of the
vertical component from the observed variations of the two horizontal
components in their own directions. Tanakadate makes a comparison of
the vertical variations of the magnetic elements calculated in the two
ways, not merely for Japan, but also for Austria-Hungary and Great
Britain. In each country he took five representative points, those for
Great Britain being the central stations of five of Rücker and
Thorpe's districts. Table XLVII. gives the mean of the five values
obtained. By method (i.) is meant the formula involving 3h/R, by
method (ii.) Tanakadate's method as explained above. H, V, D, and I
are used as defined in § 5. In the case of H and V unity represents
1[gamma].

TABLE XLVII.--Change per Kilometre of Height.

+---------+-----------------+-----------------+-----------------+
| | Great Britain. | Austria-Hungary.| Japan. |
+---------+-------+---------+-------+---------+-------+---------+
| Method. | (i.) | (ii.) | (i.) | (ii.) | (i.) | (ii.) |
+---------+-------+---------+-------+---------+-------+---------+
| H | - 8.1 | - 6.7 | -10.1 | - 8.7 | -13.9 | -14.0 |
| V | -21.2 | -19.4 | -19.0 | -18.1 | -17.1 | -17.4 |
| D (west)| .. | - 0´.04 | .. | + 0´.10 | .. | - 0´.27 |
| I | .. | - 0´.05 | .. | - 0´.06 | .. | - 0´.01 |
+---------+-------+---------+-------+---------+-------+---------+

The - sign in Table XLVII. denotes a decrease in the numerical values
of H, V and I, and a diminution in westerly declination. If we except
the case of the westerly component of force--not shown in the
table--the accordance between the results from the two methods in the
case of Japan is extraordinarily close, and there is no very marked
tendency for the one method to give larger values than the other. In
the case of Great Britain and Austria the differences between the two
sets of calculated values though not large are systematic, the 3h/R
formula invariably showing the larger reduction with altitude in both
H and V. Tanakadate was so satisfied with the accordance of the two
methods in Japan, that he employed his method to reduce all observed
Japanese values to sea-level. At a few of the highest Japanese
stations the correction thus introduced into the value of H was of
some importance, but at the great majority of the stations the
corrections were all insignificant.

Schuster's Diurnal Variation Potential.