New volumes of the Encyclopædia Britannica > Volume 30, K-MOR

462 MAGNETISM, TERRESTRIAL
where n and m denote any positive integers, m being not greater
than n. Then denoting by R the earth’s mean radius, we have
V/R=2(R/r)m+1[H%C cos ™-\+hZ sinmX)]
+ 2(r/R)w[H^-r cos m\ + h_n sin mX)];
where 2 denotes summation of m from o to n inclusive, and then
summation of n from o to co. In this equation gn, nn, g-n> ^-w
are constants, the former pair representing what are generally
termed Gaussian constants. _ .
The series with negative powers of r answers to forces whose origin
is within the earth’s surface, that with positive powers to forces
with an external origin. Gauss found that forces of this latter class
if existent were very small, and usually they have been left out of
account. o i i
There are three Gaussian constants of the first order gv, g^ hx,
five of the second order, seven of the third, and so on. The co¬
efficient of a Gaussian constant of the wth order is a spherical har¬
monic of the wth degree. It is perhaps most usual to take R as the
unit length, in which case the first order terms may be written
yi = r-2 jr/° sin l +(r/| cosX + /<’ sin X) cos Z j-.
In reality the earth is not a perfect sphere, and in the elaborate
work on this subject by J. C. Adams it is treated as a spheroid.
It is usual, however, to treat the earth as a sphere, and for sim¬
plicity we shall here treat it as such. We then have for components
of force
X= -r-1(l - /dg) towards the north,
Y=-r-1(l-aiV^ZY/cZX ,, west,
Z— - dVjdr vertically downwards.
Over the surface ?’=R, and the expressions for the components of
force are functions only of X and g. Supposing all the Gaussian
constants for any given epoch known, the above expressions would
supply the values of X, Y, and Z all over the earth s surface. To
determine the Gaussian constants for any given epoch we proceed in
the reverse direction, i.e., we equate observed values of X, Y, and Z
to the theoretical values involving g™, &c.
If we knew the value of the component forces at regularly distri¬
buted stations all over the earth’s surface, we could determine each
Gaussian constant independently of the others. Different ways of
doing this have been indicated by J. C. Adams and Schuster. Our
knowledge, however, of either Ai'ctic or Antarctic regions is scanty,
and in practice recourse must be had to methods in which the
constants are not determined independently. The consequence,
unfortunately, is that the values found for some of the less important
constants, even of the lower orders, may depend very sensibly on
how large a portion of the polar regions is excluded from the calcula¬
tion, and on the number of constants of the higher orders which are
retained. This is shown very clearly by W. G. Adams {Brit. Assoc.
Report for 1898, p. 109) in discussing the results_ found by J. C.
Adams for the two epochs 1845 and 1880, notably in the case of the
o
constant g^
W. G. Adams devotes several tables to the values of the Gaussian
constants as found by Erman-Petersen for 1829, by Gauss for 1830,
by J. C. Adams for 1845 and 1880, and by Neumayer, Schmidt,
and Fritsche for 1885. The most recent of these data for the con¬
stants of the first order are as follows :—
Table XL—Gaussian Constants of the First Order.
Constant
^ •
h\ .
Adams.
1880.
+ -316843
+ -024273
- -060300
Neumayer.
1885.
+ -315720
+ -024814
- -060258
Schmidt.
1885.
+ -317346
+ -023556
- -059842
Fritsche.
1885.
+ -31635
+ -02414
- -05914
The agreement is closer than might have been expected, consider¬
ing that the number of constants retained was different in each case,
Neumayer, for instance, neglecting all but the first twenty-four, as
against fifty retained by Adams. Even in the case of the higher
constants there are few instances of differences in sign. Still the last
two of these figures at least had better be regarded as ornamental.
It should be noticed that writers differ in the sign they give to Y,
and that, owing to a difference in his definition of a surface harmonic,
Schmidt’s values for the Gaussian constants, as given elsewhere by
himself, require multiplication by factors as explained by W. G.
Adams (Zoc. cit. pp. 134-136).
§ 24. The neglect of the second series in our original expression
for Y is, as already stated, one source of uncertainty, and a second is
the possibility that some part of the magnetic forces may not arise
from a potential. J. C. Adams made some calculations according to
which g~Jg\ is very small, but g _ g came out somewhat large.
Schmidt’s calculations led him to conclude that about one-fortieth
of the force on the average arose from a potential answering to
external forces, and that an even larger part had no potential.
This would imply that the line integral of the magnetic force round
closed areas on the earth’s surface does not in general vanish. The
physical concomitant would be vertical earth-air electric currents.
Schmidt’s final estimate of the average intensity of these currents
at the epoch 1885 appears to be 0-17 ampere per square kilometre
{Abhand. der bayer. Akad. der Wiss. Bd. xix. 1895). An earlier
paper put it at 0T {Brit. Assoc. Report for 1894, p. 570). Bauer
(T, vol. ii. p. 11), employing the same experimental data as Schmidt,
reached a similar conclusion from the differences between integrals,
taken round parallels of latitude at intervals of 5° from 60° N. to
60° S. Bauer found a very asymmetrical distribution, the current
being downwards between 5° and 45° N. and between 15° and
40° S., while elsewhere it was upwards. Yon Bezold also took
line integrals round parallels of latitude, and comparing their
values with the potential ranges as given by Schmidt’s figures,
obtained evidence favourable to the existence of vertical currents
{Sitz. k. Akad. der Wiss., Berlin, 1897, No. xviii. ; also T, vol. iii.
p. 191). Fritsche has treated the problem similarly, but has con¬
sidered a larger number of parallels, and at two separate epochs,
viz., 1842 and 1885 {Die Elements des Erdmagnetismus, p. 103).
The values of the integrals round the same parallel at the two
epochs seldom agreed closely, and sometimes differed in sign. As a
large alteration in the phenomena in the course of forty-three years
seems improbable, the most natural conclusion is that the values of
line integrals round parallels of latitude are too uncertain to afford
a satisfactory criterion as to the real existence of earth-air currents.
Similar negative results seem to flow from the calculation of line
integrals round the best surveyed areas in Europe (see von Bezold,
l.c. ; Riicker, T, vol. i. p. 77, and Nature, vol. Ivii. pp. 160 and
180 ; Liznar, Met. Zeit. for 1898, p. 175 ; and von Gyllenskold,
k. Svenska Vet. Akad. Handlingar, vol. xxvii. No. 7, 1895).
There are somewhat serious physical difficulties in the way of cur¬
rents of the size calculated by Schmidt (see Atmospheric Elec¬
tricity, Ency. Brit. vol. xxv. § 13).
§ 25. The forces arising from terms in the Gaussian potential which
depend on constants of the nth order vary as r-"-2, where r is the
distance from the earth’s centre. If F represent the jaftaence
value at sea-level of any component of force answering of ajtitude^
to Gaussian constants of the nth order, the value F + 5F
at height h is given by (F + 5F)/F = {R/(R + A)}”+2, where R is the
earth’s radius. Thus so long as A/R is small we have very approxi¬
mately 5F/F= - {n + 2)A/R. As we have seen, the terms depend¬
ing on Gaussian constants of the first order {i.e., n=\) are much
the most important; thus on the Gaussian theory we should have
approximately
5X/X = SY/Y = 5Z/Z = - 3A/R.
Thus all the components should decrease as we go upwards at the
same rate, the declination and inclination remaining unaffected.
Liznar has compared this equation with the results of his Austrian
survey {Sitz. der k. Kais. Akad. der Wiss. Wien., Math.-Nat.
Classe, Bd. cvii. Abth. ii., 1898). Subdividing his stations into
three groups according to altitude, he concluded that the westerly
component and the declination increased with the altitude, and
that while the northerly and vertical components decreased, they
did so at thrice the rate indicated by theory. These conspicuous
departures from theory have been ascribed to electrical currents in
the atmosphere, some of which would have to be horizontal.
Geitel has pointed out serious difficulties in the way of this
explanation (T, vol. iv. p. 63). Yan Ryckevorsel and van Bem-
melen found the horizontal force on the Rigi to decrease slightly
with the altitude, while the vertical force increased in somewhat
larger measure; but the differences were so small that they were
indisposed to regard their conclusions as final (T, vol. ii., 1897,
p. 76). It should be noticed that Liznar’s theoretical formula, like
the Gaussian analysis—so long as only lower order terms are re¬
tained—applies only to the earth’s field as freed from all local or
even regional disturbances ; and that presumably a good deal of
the material above sea-level in Austria is by no means non-magnetic.
According to Liznar’s views, diurnal variation phenomena should
vary sensibly with the height: if this be the case, his advocacy of
the institution of some high-level magnetic observatories is worthy
of support.
§ 26. Schuster has attempted to obtain a potential analogous to
the Gaussian potential from which the regular diurnal changes of
the magnetic elements all over the earth may be found by differ¬
entiation {Phil. Trans., A, vol. clxxx. 1889, p. 467). From the mean
summer and winter diurnal variations of the northerly and easterly
components of force during 1870 at Greenwich, Lisbon, St Peters¬
burg, and Bombay, he calculated the values of eight constants
analogous to Gaussian constants, and from considerations as to the
times of occurrence of the diurnal maxima and minima of vertica