A Treatise on Electricity and Magnetism, Vol. 2 (1873) — part 7 of 27

Since these measurements are made near the surface of the earth, the magnets are always acted on by gravity as well as by terrestrial magnetism, and since the magnets are made of steel their mag netism is partly permanent and partly induced. The permanent magnetism is altered by changes of temperature, by strong in duction, and by violent blows ; the induced magnetism varies with every variation of the external magnetic force.

The most convenient way of observing the force acting on a magnet is by making the magnet free to turn about a vertical axis. In ordinary compasses this is done by balancing the magnet on a vertical pivot. The finer the point of the pivot the smaller is the moment of the friction which interferes with the action of the magnetic force. For more refined observations the magnet is suspended by a thread composed of a silk fibre without twist, either single, or doubled on itself a sufficient number of times, and so formed into a thread of parallel fibres, each of which supports as nearly as possible an equal part of the weight. The force of torsion of such a thread is much less than that of a metal wire of equal strength, and it may be calculated in terms of the ob served azimuth of the magnet, which is not the case with the force arising from the friction of a pivot.

The suspension fibre can be raised or lowered by turning a horizontal screw which works in a fixed nut. The fibre is wound round the thread of the screw, so that when the screw is turned the suspension fibre always hangs in the same vertical line.

450.]

SUSPENSION".

89

The suspension fibre carries a small horizontal divided circle called the Torsion-circle, and a stirrup with an index, which can be placed so that the index coincides with any given division of the torsion circle. The stirrup is so shaped that the magnet bar can be fitted into it with its axis horizontal, and with any one of its four sides uppermost.

To ascertain the zero of torsion a non-magnetic body of the same weight as the magnet is placed in the stirrup, and the position of the torsion circle when in equilibrium ascertained.

The magnet itself is a piece of hard-tempered steel. According to Gauss and Weber its length ought to be at least eight times its greatest transverse dimension. This is neces sary when permanence of the direc tion of the magnetic axis within the magnet is the most important con sideration. Where promptness of motion is required the magnet should be shorter, and it may even be ad visable in observing sudden altera tions in magnetic force to use a bar magnetized transversely and sus pended with its longest dimension vertical *.

450.1 The magnet is provided with an arrangement for ascertaining its angular position. For ordinary pur poses its ends are pointed, and a divided circle is placed below the

Fig. 13.

ends, by which their positions are read oif by an eye placed in a plane through the suspension thread and the point of the needle.

For more accurate observations a plane mirror is fixed to the magnet, so that the normal to the mirror coincides as nearly as possible with the axis of magnetization. This is the method adopted by Gauss and Weber.

Another method is to attach to one end of the magnet a lens and to the other end a scale engraved on glass, the distance of the lens

• Joule, Proc. Phil. Soc., Manchester, Nov. 29, 1864.

90 MAGNETIC MEASUREMENTS. [45O.

from the scale being1 equal to tlie principal focal length of the lens. The straight line joining the zero of the scale with the optical centre of the lens ought to coincide as nearly as possible with the magnetic axis.

As these optical methods of ascertaining the angular position of suspended apparatus are of great importance in many physical researches, we shall here consider once for all their mathematical theory.

Theory of the Mirror Method.

We shall suppose that the apparatus whose angular position is to be determined is capable of revolving about a vertical axis. This axis is in general a fibre or wire by which it is suspended. The mirror should be truly plane, so that a scale of millimetres may be seen distinctly by reflexion at a distance of several metres from the mirror.

The normal through the middle of the mirror should pass through the axis of suspension, and should be accurately horizontal. We shall refer to this normal as the line of collimation of the ap paratus.

Having roughly ascertained the mean direction of the line of collimation during the experiments which are to be made, a tele scope is erected at a convenient distance in front of the mirror, and a little above the level of the mirror.

The telescope is capable of motion in a vertical plane, it is directed towards the suspension fibre just above the mirror, and a fixed mark is erected in the line of vision, at a horizontal distance from the object glass equal to twice the distance of the mirror from the object glass. The apparatus should, if possible, be so arranged that this mark is on a wall or other fixed object. In order to see the mark and the suspension fibre at the same time through the telescope, a cap may be placed over the object glass having a slit along a vertical diameter. This should be removed for the other observations. The telescope is then adjusted so that the mark is seen distinctly to coincide with the vertical wire at the focus of the telescope. A plumb-line is then adjusted so as to pass close in front of the optical centre of the object glass and to hang below the telescope. Below the telescope and just behind the plumb-line a scale of equal parts is placed so as to be bisected at right angles by the plane through the mark, the suspension-fibre, and the plumb-lino. The sum of the heights of the scale and the

450.]

THE MIRROR METHOD.

91

object glass should be equal to twice the height of the mirror from the floor. The telescope being now directed towards the mirror will see in it the reflexion of the scale. If the part of the scale where the plumb-line crosses it appears to coincide with the vertical wire of the telescope, then the line of collimation of the mirror coincides with the plane through the mark and the optical centre of the object glass. If the vertical wire coincides with any other division of the scale, the angular position of the line of collimation is to be found as follows : —

Let the plane of the paper be horizontal, and let the various points be projected on this plane. Let 0 be the centre of the object glass of the telescope, P the fixed mark, P and the vertical wire of the telescope are conjugate foci with respect to the object glass. Let M be the point where OP cuts the plane of the mirror. Let MN be the normal to the mirror ; then OMN = 6 is the angle which the line of collimation makes with the fixed plane. Let MS be a line in the plane of OM and MN, such that NMS = OMN, then S will be the part of the scale which will be seen by reflexion to coincide with the vertical wire of the telescope. Now, since

X

X

xx ---'V

Fig. 14.

MN is horizontal, the projected angles OMN and NMS in the figure are equal, and QMS =20. Hence OS = OMtan.20.

We have therefore to measure OM in terms of the divisions of the scale ; then, if s0 is the division of the scale which coincides with the plumb-line, and s the observed division,

whence 6 may be found. In measuring OM we must remember that if the mirror is of glass, silvered at the back, the virtual image of the reflecting surface is at a distance behind the front surface

92

MAGNETIC MEASUREMENTS.

[450.

of the glass = — , where t is the thickness of the glass, and //, is

the index of refraction.

We must also remember that if the line of suspension does not pass through the point of reflexion, the position of M will alter with 0. Hence, when it is possible, it is advisable to make the centre of the mirror coincide with the line of suspension.

It is also advisable, especially when large angular motions have to be observed, to make the scale in the form of a concave cylindric surface, whose axis is the line of suspension. The angles are then observed at once in circular measure without reference to a table of tangents. The scale should be carefully adjusted, so that the axis of the cylinder coincides with the suspension fibre. The numbers on the scale should always run from the one end to the other in the same direction so as to avoid negative readings. Fig. 1 5

Fig. 15.

represents the middle portion of a scale to be used with a mirror and an inverting telescope.

This method of observation is the best when the motions are slow. The observer sits at the telescope and sees the image of the scale moving to right or to left past the vertical wire of the telescope. With a clock beside him he can note the instant at which a given division of the scale passes the wire, or the division of the scale which is passing at a given tick of the clock, and he can also record the extreme limits of each oscillation.

When the motion is more rapid it becomes impossible to read the divisions of the scale except at the instants of rest at the extremities of an oscillation. A conspicuous mark may be placed at a known division of the scale, and the instant of transit of this mark may be noted.

When the apparatus is very light, and the forces variable, the motion is so prompt and swift that observation through a telescope

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 photo graphic 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

94 MAGNETIC MEASUKEMENTS. [45 2.

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 tele scope 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 z 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 I, m, n and A, /u, 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 6 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 (3 the reading of the index of the torsion circle, then a — /3 is the azimuth of the lower end of the suspension fibre.

Let y be the value of a — /3 when there is no torsion, then the moment of the force of torsion tending to diminish a will be

T(a-/3-y),

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

To determine A, fix the stirrup so that y is vertical and up wards, z to the north and so to the west, and observe the azimuth f of the line of collimation. Then remove the magnet, turn it through an angle TT about the axis of z and replace it in this inverted position, and observe the azimuth f of the line of col limation when y is downwards and x to the east,

452.] BISECTION OF MAGNETIC FOKCE. 95

f=a+f-A, (1)

r=a-|+A. (2)

Hence x = |+i(f-0. (3)

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

1— T (a— /3 — y). (4)

But if C is the observed azimuth of the line of collimation

C=a+|-A, (5)

so that the force may be written

MHsin *» sin (d - f + J- A) -T (f + A- - — 0 - y) • (6)

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

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 d — £+ 1—\ for the sine of that angle.

• If we give to /3, the reading of the torsion circle, two different values, p! and /32, and if £ and £2 are the corresponding values of £

MHsinm^-Q = r (£-£_& + &), (7)

or, if we put

" , (8)

and equation (7) becomes, dividing by Jf/Jsin m,

-^-y = 0. (9)

If we now reverse the magnet so that y is downwards, and adjust the apparatus till y is exactly vertical, and if f is the new value of the azimuth, and 5' the corresponding declination,

/(f-X + -/3-y=0> (10)

whence - = i (f+C') + i/ (C+C'-2(/3-f y)). (11)

96 MAGNETIC MEASUREMENTS. [452.

The reading of the torsion circle should now be adjusted, so that the coefficient of r may be as nearly as possible zero. For this purpose we must determine y, the value of a — (3 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 — /3 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

whose magnetic moment is - of that of the principal magnet.

Ifi

Since r remains the same, / will become m'} and if (^ and f/ are the values of ( as found by the torsion bar,

6 = iCt + fiO+i*!" (£ + &'- 2 (/3 + y)). (12)

Subtracting this equation from (11),

2(»-l)(/3 + y) = (» + ^)(CI + C1')-(l + ^,)tf+O. (13)

Having found the value of /3-fy in this way, /3, the reading of the torsion circle, should be altered till

f+f'-2(/3 + y) = 0, (14)

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

Then, since r' is a very small numerical quantity, and since its coefficient is very small, the value of the second term in the ex pression for 5 will not vary much for small errors in the values of T and y, which are the quantities whose values are least ac curately 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 5'= 8.

When great accuracy is required it is necessary to take account of the variations of 8 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 r], if are the observed azimuths of the second magnet corresponding to f and f ', and if 8 and 8' are the corresponding values of 8, then 8'-8 = rj'-r?. (15)

Hence, to find the value of 8 we must add to (11) a correction

i (')-•?')•

The declination at the time of the first observation is therefore

8 = 4(C+r+ ^-770 + 4/^+^-2/3-2^. (16)

453-] OBSERVATION OP DEFLEXION. 97

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

^ = A + i(f-r)-H^-^Hi^(f-r-f2A-7r). (17)

By repeating the experiments with the bar on its two edges, so that the axis of OB is vertically upwards and downwards, we can find the value of m. If the axis of collimation is capable of ad justment 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 //, 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 r from the centre of the magnet in the positive direction of the axis of the magnet, is ^ = 2— (I)

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 It,

=2(l + 2§ + sg + &c.). 00

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

JL)+fcc., (3)

where Alt 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

• See a Paper on 'Imperfect Inversion,' by W. Swan. Trans. R. S. Edin., vol. xxi (1855), p. 349.

VOL. TT. H

98 MAGNETIC MEASUREMENTS. [453-

the west. The resultant force will make an angle 0 with the magnetic meridian, measured towards the west, and such that

(4)

Hence, to determine -~= we proceed as follows : — JdL

The direction of the magnetic north having been ascertained, a magnet, whose dimensions should not be too great, is suspended as in the former experiments, and the deflecting magnet M is placed so that its centre is at a distance r from that of the sus pended magnet, in the same horizontal plane, and due magnetic east.

The axis of M is carefully adjusted so as to be horizontal and in the direction of r.

The suspended magnet is observed before M is brought near and also after it is placed in position. If 0 is the observed deflexion, we have, if we use the approximate formula ( 1 ),

f=^tau*; (5)

or, if we use the formula (3), •.-•••. \ JrHan^l + ^i+^+fec. (6)

Here we must bear in mind that though the deflexion 0 can be observed with great accuracy, the distance r between the centres of the magnets is a quantity which cannot be precisely deter mined, unless both magnets are fixed and their centres defined by marks.

This difficulty is overcome thus :

The magnet M is placed on a divided scale which extends east and west on both sides of the suspended magnet. The middle point between the ends of M is reckoned the centre of the magnet. This point may be marked on the magnet and its position observed on the scale, or the positions of the ends may be observed and the arithmetic mean taken. Call this Sj, and let the line of the suspension fibre of the suspended magnet when produced cut the scale at *0, then r1 = s1 — s0) where ^ is known accurately and s0 ap proximately. Let 01 be the deflexion observed in this position of M.

Now reverse M, that is, place it on the scale with its ends reversed, then ^ will be the same, but M and Alt A3, &c. will have their signs changed, so that if 02 is ^ne deflexion,

• I r,»tan 9, = 1 -A, ± + J,± -&c. (7)

454-] DEFLEXION OBSERVATIONS. 99

Taking the arithmetical mean of (6) and (7),

i ^(tan^-tanfy = 1+^72 +^4^ + &c. (8)

Now remove M to the west side of the suspended magnet, and place it with its centre at the point marked 2<$0 — s on the scale. Let the deflexion when the axis is in the first position be 03, and when it is in the second 04, then, as before,

2

Let us suppose that the true position of the centre of the sus pended magnet is not SQ but <?0 -f or, then

(10)

and (V +, 2») = ,»(!. + 'l^ + &c.); (11)

O

and since -^ may be neglected if the measurements are .carefully

made, we are sure that we may take the arithmetical mean of rLn and r2n for rn.

Hence, taking the arithmetical mean of (8) and (9),

--^ or, making

= 1 + A2~ +&c., (12)

• (tan Ol — tan 62 + tan 03 — tan 04) = D, (13)

454.] We may now regard D and r as capable of exact deter mination.

The quantity A2 can in no case exceed 2^2, where L is half the length of the magnet, so that when r is considerable compared with L we may neglect the term in A2 and determine the ratio of H to M at once. We cannot, however, assume that A2 is equal to 2i/2, for it may be less, and may even be negative for a magnet whose largest dimensions are transverse to the axis. The term in A±, and all higher terms, may safely be neglected.

To eliminate A2, repeat the experiment, using distances rlt ra, ?*3, &c., and let the values of D be J)19 D2, #3, &c., then

-2M(l , 4

2~~iT^ + ^

&c. &c.

II 2

100 MAGNETIC MEASUREMENTS. [454-

If we suppose that the probable errors of these equations are equal, as they will be if they depend on the determination of D only, and if there is no uncertainty about r, then, by multiplying each equation by r3 and adding the results, we obtain one equation, and by multiplying each equation by r5 and adding we obtain another, according to the general rule in the theory of the com bination of fallible measures when the probable error of each equation is supposed the same.

Let us write

2(Vr-*) for AT3 + -02V3 + A^f 3 + &c., and use similar expressions for the sums of other groups of symbols, then the two resultant equations may be written

*} = (2 (r-&) + 4 2

O TUT

2 (J)r~5) = -g- (2 (*-«) + A2 2

whence

1 W

-=- 2 /-6 2r-10~2/-82 = 2 Z>r

and 4>{2 (D?-3) 2 (r10)-2 (Dr5) 2 (*-8)}

= 2 (Dr-B) 2 (r-«)-2 (Dr~*) 2 (r-8).

The value of A2 derived from these equations ought to be less than half the square of the length of the magnet M. If it is not we may suspect some error in the observations. This method of observation and reduction was given by Gauss in the ( First Report of the Magnetic Association/

When the observer can make only two series of experiments at

2M

distances r± and r2, the value of -=- derived from these experi

ments is

If 5Z)X and bD2 are the actual errors of the observed deflexions ^ and _Z)2, the actual error of the calculated result Q will be

If we suppose the errors 8^ and bD2 to be independent, and that the probable value of either is SD, then the probable value of the error in the calculated value of Q will be 5 Q, where

455-1 METHODS OF TANGENTS AND SINES. 101

If we suppose that one of these distances, say the sinaHar,; ijs- given, the value of the greater distance may be determined so as to make b Q a minimum. This condition leads to an equation of the fifth degree in rf^ which has only one real root greater than r22. From this the best value of ^ is found to be rx = 1.3189/2*.

If one observation only is taken the best distance is when

bD r-lr -— = x/3 — , D ° r

where b D is the probable error of a measurement of deflexion, and br is the probable error of a measurement of distance.

Method of Sines.

455.] The method which we have just described may be called the Method of Tangents, because the tangent of the deflexion is a measure of the magnetic force.

If the line rl5 instead of being measured east or west, is adjusted till it is at right angles with the axis of the deflected magnet, then R is the same as before, but in order that the suspended magnet may remain perpendicular to r, the resolved part of the force H in the direction of r must be equal and opposite to R. Hence, if 0 is the deflexion, R — Hsm 0.

This method is called the Method of Sines. It can be applied only when R is less than H.

In the Kew portable apparatus this method is employed. The suspended magnet hangs from a part of the apparatus which revolves along with the telescope and the arm for the deflecting magnet, and the rotation of the whole is measured on the azimuth circle.

The apparatus is first adjusted so that the axis of the telescope coincides with the mean position of the line of collimation of the magnet in its undisturbed state. If the magnet is vibrating, the true azimuth of magnetic north is found by observing the ex tremities of the oscillation of the transparent scale and making the proper correction of the reading of the azimuth circle.

The deflecting magnet is then placed upon a straight rod which passes through the axis of the revolving apparatus at right angles to the axis of the telescope, and is adjusted so that the axis of the deflecting magnet is in a line passing through the centre of the suspended magnet.

The whole of the revolving apparatus is then moved till the line

• See Airy's Magnetism.

102 MAGNETIC MEASUREMENTS. [45$.

of coilimation of the suspended magnet again coincides with the axis of the telescope, and the new azimuth reading is corrected, if necessary, by the mean of the scale readings at the extremities of an oscillation.

The difference of the corrected azimuths gives the deflexion, after which we proceed as in the method of tangents, except that in the expression for D we put sin & instead of tan 6.

In this method there is no correction for the torsion of the sus pending fibre, since the relative position of the fibre, telescope, and magnet is the same at every observation.

The axes of the two magnets remain always at right angles in this method, so that the correction for length can be more ac curately made.

456.] Having thus measured the ratio of the moment of the deflecting magnet to the horizontal component of terrestrial mag netism, we have next to find the product of these quantities, by determining the moment of the couple with which terrestrial mag netism tends to turn the same magnet when its axis is deflected from the magnetic meridian.

There are two methods of making this measurement, the dy namical, in which the time of vibration of the magnet under the action of terrestrial magnetism is observed, and the statical, in which the magnet is kept in equilibrium between a measurable statical couple and the magnetic force.

The dynamical method requires simpler apparatus and is more accurate for absolute measurements, but takes up a considerable time, the statical method admits of almost instantaneous measure ment, and is therefore useful in tracing the changes of the intensity of the magnetic force, but it requires more delicate apparatus, and is not so accurate for absolute measurement.

Method of Vibrations.

The magnet is suspended with its magnetic axis horizontal, and is set in vibration in small arcs. The vibrations are observed by means of any of the methods already described.

A point on the scale is chosen corresponding to the middle of the arc of vibration. The instant of passage through this point of the scale in the positive direction is observed. If there is suffi cient time before the return of the magnet to the same point, the instant of passage through the point in the negative direction is also observed, and the process is continued till n+I positive and

456.] TIME OF VIBKATION. 103

n negative passages have been observed. If the vibrations are too rapid to allow of every consecutive passage being observed, every third or every fifth passage is observed, care being taken that the observed passages are alternately positive and negative.

Let the observed times of passage be T1} T2, T2n+1, then if we put I 4 + y + y 4 &c.

then Tn+1 is the mean time of the positive passages, and ought to agree with T'n+v the mean time of the negative passages, if the point has been properly chosen. The mean of these results is to be taken as the mean time of the middle passage.

After a large number of vibrations have taken place, but before the vibrations have ceased to be distinct and regular, the observer makes another series of observations, from which he deduces the mean time of the middle passage of the second series.

By calculating the period of vibration either from the first series of observations or from the second, he ought to be able to be certain of the number of whole vibrations which have taken place in the interval between the time of middle passage in the two series. Dividing the interval between the mean times of middle passage in the two series by this number of vibrations, the mean time of vibration is obtained.

The observed time of vibration is then to be reduced to the time of vibration in infinitely small arcs by a formula of the same kind as that used in pendulum observations, and if the vibrations are found to diminish rapidly in amplitude, there is another cor rection for resistance, see Art. 740. These corrections, however, are very small when the magnet hangs by a fibre, and when the arc of vibration is only a few degrees.

The equation of motion of the magnet is

•  =  0 
  

where 0 is the angle between the magnetic axis and the direction of the force H, A is the moment of inertia of the magnet and suspended apparatus, M is the magnetic moment of the magnet, H the intensity of the horizontal magnetic force, and MHr' the coefficient of torsion : / is determined as in Art. 452, and is a very small quantity. The value of 0 for equilibrium is

T "y 00 = - - T 5 a very small angle,

104 MAGNETIC MEASUREMENTS. [457-

and the solution of the equation for small values of the amplitude,

C is f t \

0 = Ccos (2 TT -^ 4- a) + 00,

where T is the periodic time, and C the amplitude, and

yr2

whence we find the value of MH9

Here T is the time of a complete vibration determined from observation. A, the moment of inertia, is found once for all for the magnet, either by weighing and measuring it if it is of a regular figure, or by a dynamical process of comparison with a body whose moment of inertia is known.

Combining this value of Mil with that of -~ formerly obtained, we get Jp

and //*

457.] We have supposed that //and M continue constant during the two series of experiments. The fluctuations of // may be ascertained by simultaneous observations of the bifilar magnet ometer to be presently described, and if the magnet has been in use for some time, and is not exposed during the experiments to changes of temperature or to concussion, the part of M which de pends on permanent magnetism may be assumed to be constant. All steel magnets, however, are capable of induced magnetism depending on the action of external magnetic force.

Now the magnet when employed in the deflexion experiments is placed with its axis east and west, so that the action of ter restrial magnetism is transverse to the magnet, and does not tend to increase or diminish M. When the magnet is made to vibrate, its axis is north and south, so that the action of terrestrial mag netism tends to magnetize it in the direction of the axis, and therefore to increase its magnetic moment by a quantity Jc //, where k is a coefficient to be found by experiments on the magnet.

There are two ways in which this source of error may be avoided without calculating Jc, the experiments being arranged so that the magnet shall be in the same condition when employed in deflecting another magnet and when itself swinging.

457-] ELIMINATION OF INDUCTION. 105