451.]

METHODS OF OBSERVATION.

93

would be useless. In this case the observer looks at the scale directly, and observes the motions of the image of the vertical wire thrown on the scale by a lamp.

It is manifest that since the image of the scale reflected by the mirror and refracted by the object glass coincides with the vertical wire, the image of the vertical wire, if sufficiently illuminated, will coincide with the scale. To observe this the room is darkened, and the concentrated rays of a lamp are thrown on the vertical wire towards the object glass. A bright patch of light crossed by the shadow of the wire is seen on the scale. Its motions can be followed by the eye, and the division of the scale at which it comes to rest can be fixed on by the eye and read off at leisure. If it be desired to note the instant of the passage of the bright spot past a given point on the scale, a pin or a bright metal wire may be placed there so as to flash out at the time of passage.

By substituting a small hole in a diaphragm for the cross wire the image becomes a small illuminated dot moving to right or left on the scale, and by substituting for the scale a cylinder revolving by clock work about a horizontal axis and covered with photographic paper, the spot of light traces out a curve which can be afterwards rendered visible. Each abscissa of this curve corresponds to a particular time, and the ordinate indicates the angular position of the mirror at that time. In this way an automatic system of continuous registration of all the elements of terrestrial magnetism has been established at Kew and other observatories.

In some cases the telescope is dispensed with, a vertical wire is illuminated by a lamp placed behind it, and the mirror is a concave one, which forms the image of the wire on the scale as a dark line across a patch of light.

451.] In the Kew portable apparatus, the magnet is made in the form of a tube, having at one end a lens, and at the other a glass scale, so adjusted as to be at the principal focus of the lens. Light is admitted from behind the scale, and after passing through the lens it is viewed by means of a telescope.

Since the scale is at the principal focus of the lens, rays from any division of the scale emerge from the lens parallel, and if the telescope is adjusted for celestial objects, it will shew the scale in optical coincidence with the cross wires of the telescope. If a given division of the scale coincides with the intersection of the cross wires, then the line joining that division with the optical centre of the lens must be parallel to the line of collimation of

the telescope. By fixing the magnet and moving the telescope, we may ascertain the angular value of the divisions of the scale, and then, when the magnet is suspended and the position of the telescope known, we may determine the position of the magnet at any instant by reading off the division of the scale which coincides with the cross wires.

The telescope is supported on an arm which is centred in the line of the suspension fibre, and the position of the telescope is read off by verniers on the azimuth circle of the instrument.

This arrangement is suitable for a small portable magnetometer in which the whole apparatus is supported on one tripod, and in which the oscillations due to accidental disturbances rapidly subside.

Determination of the Direction of the Axis of the Magnet, and of the Direction of Terrestrial Magnetism.

452.] Let a system of axes be drawn in the magnet, of which the axis of is in the direction of the length of the bar, and x and y perpendicular to the sides of the bar supposed a parallelepiped.

Let l, m, n and λ, μ, v be the angles which the magnetic axis and the line of collimation make with these axes respectively.

Let M be the magnetic moment of the magnet, let H be the horizontal component of terrestrial magnetism, let Z be the vertical component, and let ò be the azimuth in which H acts, reckoned from the north towards the west.

Let be the observed azimuth of the line of collimation, let a be the azimuth of the stirrup, and ẞ the reading of the index of the torsion circle, then a-ẞ is the azimuth of the lower end of the suspension fibre.

γ

Let be the value of a-ẞ when there is no torsion, then the moment of the force of torsion tending to diminish a will be

T (a-B-y),

where is a coefficient of torsion depending on the nature of the fibre.

To determine A, fix the stirrup so that y is vertical and upwards, z to the north and a to the west, and observe the azimuth

of the line of collimation. Then remove the magnet, turn it through an angle about the axis of z and replace it in this inverted position, and observe the azimuth of the line of collimation when y is downwards and x to the east,

Next, hang the stirrup to the suspension fibre, and place the magnet in it, adjusting it carefully so that y may be vertical and upwards, then the moment of the force tending to increase a is

But if is the observed azimuth of the line of collimation

(4)

(5)

(6)

π

MH sin m sin (ò —§+1—λ)—r (5+λ— — — ß − y)·

2

When the apparatus is in equilibrium this quantity is zero for a particular value of §.

When the apparatus never comes to rest, but must be observed in a state of vibration, the value of corresponding to the position of equilibrium may be calculated by a method which will be described in Art. 735.

When the force of torsion is small compared with the moment of the magnetic force, we may put 8-+-λ for the sine of that angle.

If we give to ẞ, the reading of the torsion circle, two different values, ß1 and ẞ2, and if § and §1⁄2 are the corresponding values of (, MH sin m (-2) = T (51-52-B1+ B2),

or, if we put

(7)

(9)

and equation (7) becomes, dividing by MH sin m,

If we now reverse the magnet so that y is downwards, and adjust the apparatus till y is exactly vertical, and if is the new value of the azimuth, and d' the corresponding declination,

8′ −5′−1+λ—r′ (5′ —λ + 1 −ß−v) = 0,

(10)

The reading of the torsion circle should now be adjusted, so that the coefficient of 7' may be as nearly as possible zero. For this purpose we must determine y, the value of a-ß when there is no torsion. This may be done by placing a non-magnetic bar of the same weight as the magnet in the stirrup, and determining a—ß when there is equilibrium. Since is small, great accuracy is not required. Another method is to use a torsion bar of the same weight as the magnet, containing within it a very small magnet

1

whose magnetic moment is of that of the principal magnet.

Since

n

remains the same, 7' will become r', and if 1 and § are

the values of as found by the torsion bar,

8 = 1 (51 + $1′ ) + 1⁄2 n r′ (§1 + Ší − 2 (B+y)).

Subtracting this equation from (11),

2(n-1) (B+y)= (n + — ) (5 +5') − (1 + 1 ) (5+5).

(12)

(13)

Having found the value of ẞ+y in this way, ß, the reading of the torsion circle, should be altered till

5+5'-2 (B+ y) = 0,

as nearly as possible in the ordinary position of the apparatus.

(14)

Then, since is a very small numerical quantity, and since its coefficient is very small, the value of the second term in the expression for d will not vary much for small errors in the values of and y, which are the quantities whose values are least accurately known.

The value of 8, the magnetic declination, may be found in this way with considerable accuracy, provided it remains constant during the experiments, so that we may assume d'= d.

When great accuracy is required it is necessary to take account of the variations of ò during the experiment. For this purpose observations of another suspended magnet should be made at the same instants that the different values of are observed, and if n, n' are the observed azimuths of the second magnet corresponding to and ', and if ò and d' are the corresponding values of ô, then

8-d= n-n.

Hence, to find the value of d we must add to (11) a correction

(n−n).

The declination at the time of the first observation is therefore

8 = $ (5+ 5′ + n − n ' ) + § 7′ ( Š + 5′ — 2 ß−2y).

(15)

(16)

453.]

OBSERVATION OF DEFLEXION.

97

To find the direction of the magnetic axis within the magnet subtract (10) from (9) and add (15),

-

-

(17)

7 = λ + } (5 — 5') — § (n − n' ́) + § 7′ (5—5′ + 2λ — π). By repeating the experiments with the bar on its two edges, so that the axis of x is vertically upwards and downwards, we can find the value of m. If the axis of collimation is capable of adjustment it ought to be made to coincide with the magnetic axis as nearly as possible, so that the error arising from the magnet not being exactly inverted may be as small as possible *.

On the Measurement of Magnetic Forces.

453.] The most important measurements of magnetic force are those which determine M, the magnetic moment of a magnet, and H, the intensity of the horizontal component of terrestrial magnetism. This is generally done by combining the results of two experiments, one of which determines the ratio and the other the product of these two quantities.

The intensity of the magnetic force due to an infinitely small magnet whose magnetic moment is M, at a point distant from the centre of the magnet in the positive direction of the axis of the magnet, is

R = 2

M
2.3

(1)

and is in the direction of r. If the magnet is of finite size but spherical, and magnetized uniformly in the direction of its axis, this value of the force will still be exact. If the magnet is a solenoidal bar magnet of length 2 L,

If the magnet be of any kind, provided its dimensions are all small compared with r,

M 7.3

1

R = 2 (1 + 41 = + 41⁄2 ~—-2) + &c.,

(3)

where A1, A2, &c. are coefficients depending on the distribution of the magnetization of the bar.

Let H be the intensity of the horizontal part of terrestrial magnetism at any place. H is directed towards magnetic north. Let r be measured towards magnetic west, then the magnetic force at the extremity of r will be H towards the north and R towards Trans. R. S. Edin.,

See a Paper on 'Imperfect Inversion,' by W. Swan. vol. xxi (1855), p. 349.