force. The most convenient instrument for continuous observations is the vertical force magnetometer, which is simply a magnet balanced on knife edges so as to be in stable equilibrium with its magnetic axis nearly horizontal.

If Z is the vertical component of the magnetic force, M the magnetic moment, and the small angle which the magnetic axis makes with the horizon

MZ = mga cos (a-0),

where m is the mass of the magnet, g the force of gravity, a the distance of the centre of gravity from the axis of suspension, and a the angle which the plane through the axis and the centre of gravity makes with the magnetic axis.

Hence, for the small variation of vertical force 82, there will be a variation of the angular position of the magnet 60 such that

M8Z mga sin (a-0) 80.

In practice this instrument is not used to determine the absolute value of the vertical force, but only to register its small variations. For this purpose it is sufficient to know the absolute value of Z dz when 0 = 0, and the value of do

The value of Z, when the horizontal force and the dip are known, is found from the equation Z = Htan 0, where 6, is the dip and H the horizontal force.

To find the deflexion due to a given variation of Z, take a magnet and place it with its axis east and west, and with its centre at a known distance r1 east or west from the declinometer, as in experiments on deflexion, and let the tangent of deflexion be D1.

Then place it with its axis vertical and with its centre at a distance r2 above or below the centre of the vertical force magnetometer, and let the tangent of the deflexion produced in the magnetometer be D. Then, if the moment of the deflecting magnet is M,

Hence

3

M = Hr123D1

=

do

The actual value of the vertical force at any instant is

where Z is the value of Z when 0 = 0.

For continuous observations of the variations of magnetic force

464.]

VERTICAL FORCE.

119

at a fixed observatory the Unifilar Declinometer, the Bifilar Horizontal Force Magnetometer, and the Balance Vertical Force Magnetometer are the most convenient instruments.

At several observatories photographic traces are now produced on prepared paper moved by clock work, so that a continuous record of the indications of the three instruments at every instant is formed. These traces indicate the variation of the three rectangular components of the force from their standard values. The declinometer gives the force towards mean magnetic west, the bifilar magnetometer gives the variation of the force towards magnetic north, and the balance magnetometer gives the variation of the vertical force. The standard values of these forces, or their values when these instruments indicate their several zeros, are deduced by frequent observations of the absolute declination, horizontal force, and dip.

CHAPTER VIII.

ON TERRESTRIAL MAGNETISM.

465.] OUR knowledge of Terrestrial Magnetism is derived from the study of the distribution of magnetic force on the earth's surface at any one time, and of the changes in that distribution at different times.

The magnetic force at any one place and time is known when its three coordinates are known. These coordinates may be given in the form of the declination or azimuth of the force, the dip or inclination to the horizon, and the total intensity.

The most convenient method, however, for investigating the general distribution of magnetic force on the earth's surface is to consider the magnitudes of the three components of the force,

XH cos 8, directed due north,

Y H sind, directed due west,

Z = H tan 6, directed vertically downwards,

where H denotes the horizontal force, 8 the declination, and the dip.

(1)

If is the magnetic potential at the earth's surface, and if we consider the earth a sphere of radius a, then

where

is the latitude, and λ the longitude, and the distance

from the centre of the earth.

A knowledge of V over the surface of the earth may be obtained from the observations of horizontal force alone as follows.

Let Vo be the value of at the true north pole, then, taking the line-integral along any meridian, we find,

for the value of the potential on that meridian at latitude 7.

(3)

466.]

MAGNETIC SURVEY.

121

Thus the potential may be found for any point on the earth's surface provided we know the value of X, the northerly component at every point, and V。, the value of V at the pole.

Since the forces depend not on the absolute value of V but on its derivatives, it is not necessary to fix any particular value for Vo

The value of V at any point may be ascertained if we know the value of X along any given meridian, and also that of Y over the whole surface.

Let

K = af 'xdl + Vo

(4)

where the integration is performed along the given meridian from the pole to the parallel 7, then

Ꮴ
V = + afˆ Y cos i dλ,

λο

l

(5)

where the integration is performed along the parallel from the given meridian to the required point.

These methods imply that a complete magnetic survey of the earth's surface has been made, so that the values of X or of Y or of both are known for every point of the earth's surface at a given epoch. What we actually know are the magnetic components at a certain number of stations. In the civilized parts of the earth these stations are comparatively numerous; in other places there are large tracts of the earth's surface about which we have no data.

Magnetic Survey.

466.] Let us suppose that in a country of moderate size, whose greatest dimensions are a few hundred miles, observations of the declination and the horizontal force have been taken at a considerable number of stations distributed fairly over the country. Within this district we may suppose the value of V to be represented with sufficient accuracy by the formula

whence

V = √。+ a(A1l + A1⁄2 λ + 1⁄2 B1 l2 + B2l dλ + † B3 λ2 +&c.), (6)
X = 41+B1l+B2 λ,

(7)

(8)

Let there ben stations whose latitudes are 1, 2, ... &c. and longitudes A1, A2, &c., and let X and Y be found for each station.

and

may be called the latitude and longitude of the central station. Let

then X and Y are the values of X and Y at the imaginary central station, then

X = X2+B1 (l—b) + B2 (λ —λo),

2

Y cos l = Y cos + B2 (1 − 1) + B3 (λ—λo).

(11)

(12)

We have n equations of the form of (11) and n of the form (12). If we denote the probable error in the determination of X by έ, and that of Y cos / by n, then we may calculate έ and the supposition that they arise from errors of observation of H and d.

Let the probable error of II be h, and that of 8, d, then since d X = cos ò . d H – H sin ô . đồn

If the variations of X and Y from their values as given by equations of the form (11) and (12) considerably exceed the probable errors of observation, we may conclude that they are due to local attractions, and then we have no reason to give the ratio of έ to ŋ any other value than unity.

According to the method of least squares we multiply the equations of the form (11) by n, and those of the form (12) by έ to make their probable error the same. We then multiply each equation by the coefficient of one of the unknown quantities B1, B2, or B, and add the results, thus obtaining three equations from which to find B1, B, and B ̧.

By calculating B1, B, and B ̧, and substituting in equations (11) and (12), we can obtain the values of X and Y at any point within the limits of the survey free from the local disturbances