A Treatise on Electricity and Magnetism, Vol. 2 (1873) — part 8 of 27

We may place the deflecting magnet with its axis pointing north, at a distance r from the centre of the suspended magnet, the line r making an angle whose cosine is /J with the magnetic meridian. The action of the deflecting magnet on the suspended one is then at right angles to its own direction, and is equal to

Here M is the magnetic moment when the axis points north, as in the experiment of vibration, so that no correction has to be made for induction.

This method, however, is extremely difficult, owing to the large errors which would be introduced by a slight displacement of the deflecting magnet, and as the correction by reversing the deflecting magnet is not applicable here, this method is not to be followed except when the object is to determine the coefficient of induction.

The following method, in which the magnet while vibrating is freed from the inductive action of terrestrial magnetism, is due to Dr. J. P. Joule *.

Two magnets are prepared whose magnetic moments are as nearly equal as possible. In the deflexion experiments these mag nets are used separately, or they may be placed simultaneously on opposite sides of the suspended magnet to produce a greater deflexion. In these experiments the inductive force of terrestrial magnetism is transverse to the axis.

Let one of these magnets be suspended, and let the other be placed parallel to it with its centre exactly below that of the sus pended magnet, and with its axis in the same direction. The force which the fixed magnet exerts on the suspended one is in the opposite direction from that of terrestrial magnetism. If the fixed magnet be gradually brought nearer to the suspended one the time of vibration will increase, till at a certain point the equilibrium will cease to be stable, and beyond this point the suspended magnet will make oscillations in the reverse position. By experimenting in this way a position of the fixed magnet is found at which it exactly neutralizes the effect of terrestrial magnetism on the sus pended one. The two magnets are fastened together so as to be parallel, with their axes turned the same way, and at the distance just found by experiment. They are then suspended in the usual way and made to vibrate together through small arcs.

• Proc. Phil. S., Manchester, March 19, 1867.

106 MAGNETIC MEASUREMENTS. [45 8.

The lower magnet exactly neutralizes the effect of terrestrial magnetism on the upper one, and since the magnets are of equal moment, the upper one neutralizes the inductive action of the earth on the lower one.

The value of M is therefore the same in the experiment of vibration as in the experiment of deflexion, and no correction for induction is required.

458.] The most accurate method of ascertaining the intensity of the horizontal magnetic force is that which we have just described. The whole series of experiments, however, cannot be performed with sufficient accuracy in much less than an hour, so that any changes in the intensity which take place in periods of a few minutes would escape observation. Hence a different method is required for ob serving the intensity of the magnetic force at any instant.

The statical method consists in deflecting the magnet by means of a statical couple acting in a horizontal plane. If L be the moment of this couple, M the magnetic moment of the magnet, // the horizontal component of terrestrial magnetism, and 0 the deflexion, M H sin 0 = L.

Hence, if L is known in terms of 0, MH can be found.

The couple L may be generated in two ways, by the torsional elasticity of a wire, as in the ordinary torsion balance, or by the weight of the suspended apparatus, as in the bifilar suspension.

In the torsion balance the magnet is fastened to the end of a vertical wire, the upper end of which can be turned round, and its rotation measured by means of a torsion circle.

We have then

X, = r(a — a0 — 6) = Mil sin 6.

Here a0 is the value of the reading of the torsion circle when the axis of the magnet coincides with the magnetic meridian, and a is the actual reading. If the torsion circle is turned so as to bring the magnet nearly perpendicular to the magnetic meridian, so that

e = ~tf, then r(a-a0- + 00

or

By observing 0', the deflexion of the magnet when in equilibrium, we can calculate Mil provided we know r.

If we only wish to know the relative value of H at different times it is not necessary to know either M or T.

We may easily determine T in absolute measure by suspending

459-] BIFILAB SUSPENSION. 107

a non-magnetic body from the same wire and observing its time of oscillation, then if A is the moment of inertia of this body, and T the time of a complete vibration,

The chief objection to the use of the torsion balance is that the zero-reading a0 is liable to change. Under the constant twisting force, arising from the tendency of the magnet to turn to the north, the wire gradually acquires a permanent twist, so that it becomes necessary to determine the zero-reading of the torsion circle afresh at short intervals of time.

Bifilar Suspension.

459.] The method of suspending the magnet by two wires or fibres was introduced by Gauss and Weber. As the bifilar sus pension is used in many electrical instruments, we shall investigate it more in detail. The general appearance of the suspension is shewn in Fig. 16, and Fig. 17 represents the projection of the wires on a horizontal plane.

AB and A'B' are the projections of the two wires.

AA and BB' are the lines joining the upper and the lower ends of the wires.

a and b are the lengths of these lines.

a and /3 their azimuths.

TFand W the vertical components of the tensions of the wires.

Q and Q' their horizontal components.

h the vertical distance between AA and BB'.

The forces which act on the magnet are — its weight, the couple arising from terrestrial magnetism, the torsion of the wires (if any) and their tensions. Of these the effects of magnetism and of torsion are of the nature of couples. Hence the resultant of the tensions must consist of a vertical force, equal to the weight of the magnet, together with a couple. The resultant of the vertical components of the tensions is therefore along the line whose pro jection is 0, the intersection of A A and BB', and either of these lines is divided in 0 in the ratio of W to W.

The horizontal components of the tensions form a couple, and are therefore equal in magnitude and parallel in direction. Calling either of them Q, the moment of the couple which they form is

L=Q.PF, (1)

where PP7 is the distance between the parallel lines AB and AB'.

108

MAGNETIC MEASUREMENTS.

[459-

To find the value of L we have the equations of moments

Qh = W. AB = Jr. AK> (2)

and the geometrical equation

(AB + A'ff) PPf = ab sin (a- ft), ( 3)

whence we obtain,

ab WW

1=

W+ W

r, sin(a-/3).

Fig. 16.

Fig. 17.

(4)

If m is the mass of the suspended apparatus, and g the intensity

of gravity, w+ W' = mg. (5)

If we also write W— W — nmg> (6)

L — - (i—.n?)m.ff-jr sin (a— ft).

we find L — - (1 —nz}mff — sin (a — /3V (7)

The value of L is therefore a maximum with respect to n when n

459-] BIFILAR SUSPENSION. 109

is zero, that is, when the weight of the suspended mass is equally borne by the two wires.

We may adjust the tensions of the wires to equality by observing1 the time of vibration, and making it a minimum, or we may obtain a self-acting adjustment by attaching the ends of the wires, as in Fig. 16, to a pulley, which turns on its axis till the tensions are equal.

The distance of the upper ends of the suspension wires is re gulated by means of two other pullies. The distance between the lower ends of the wires is also capable of adjustment.

By this adjustment of the tension, the couple arising from the tensions of the wires becomes

T I ab . .

L = - -j- mg sin (a— -/3).

The moment of the couple arising from the torsion of the wires is of the form T (y—p\

where r is the sum of the coefficients of torsion of the wires.

The wires ought to be without torsion when a = ft, we may then make y — a.

The moment of the couple arising from the horizontal magnetic force is of the form

MS BIU (3 — 0),

where 8 is the magnetic declination, and 0 is the azimuth of the axis of the magnet. We shall avoid the introduction of unnecessary symbols without sacrificing generality if we assume that the axis of the magnet is parallel to £JB', or that /3 = 0. The equation of motion then becomes

4--j72= MHsw(b — 0} + - ^-^sin(a — 0) + r(a-0). (8)

There are three principal positions of this apparatus.

(1) When a is nearly equal to 8. If T^ is the time of a complete oscillation in this position, then

47r2^ lab

-yrr- = l-fi>"ff+T + MH. (9)

(2) When a is nearly equal to 8 + 77. If T2 is the time of a complete oscillation in this position, the north end of the magnet being now turned towards the south,

1 ab

^-jrWff + T-MH. (10)

The quantity on the right-hand of this equation may be made

130 MAGNETIC MEASUREMENTS. [459.

as small as we please by diminishing a or £, but it must not be made negative, or the equilibrium of the magnet will become un stable. The magnet in this position forms an instrument by which small variations in the direction of the magnetic force may be rendered sensible.

For when 5—0 is nearly equal to TT, sin (8 — 0) is nearly equal to 6 — by and we find

(8-a). (11)

= a-

7

l ah 71* rr

• ~j-mg--T —MH

4 fl

By diminishing the denominator of the fraction in the last term we may make the variation of 0 very large compared with that of 8. We should notice that the coefficient of 8 in this expression is negative, so that when the direction of the magnetic force turns in one direction the magnet turns in the opposite direction.

(3) In the third position the upper part of the suspension- apparatus is turned round till the axis of the magnet is nearly perpendicular to the magnetic meridian.

If we make

0-8=|+0/, and a-6 = p-P, (12)

the equation of motion may be written

(/:J-0'). (13) If there is equilibrium when //= EQ and 0'= 0,

= 0, (14)

and if H is the value of the horizontal force corresponding to a small angle 0/, x ^

• -j- mg cos /3 -|- T \

'--~ -

In order that the magnet may be in stable equilibrium it is necessary that the numerator of the fraction in the second member should be positive, but the more nearly it approaches zero, the more sensitive will be the instrument in indicating changes in the value of the intensity of the horizontal component of terrestrial magnetism.

The statical method of estimating the intensity of the force depends upon the action of an instrument which of itself assumes

46 1. J DTP. Ill

different positions of equilibrium for different values of the force. Hence, by means of a mirror attached to the magnet and throwing1 a spot of light upon a photographic surface moved by clockwork, a curve may be traced, from which the intensity of the force at any instant may be determined according to a scale, which we may for the present consider an arbitrary one.

460.] In an observatory, where a continuous system of regis tration of declination and intensity is kept up either by eye ob servation or by the automatic photographic method, the absolute values of the declination and of the intensity, as well as the position and moment of the magnetic axis of a magnet, may be determined to a greater degree of accuracy.

For the declinometer gives the declination at every instant affected by a constant error, and the bifilar magnetometer gives the intensity at every instant multiplied by a constant coefficient. In the ex periments we substitute for b, 8 + 80 where 8' is the reading of the declinometer at the given instant, and 80 is the unknown but constant error, so that 8' + 80 is the true declination at that instant.

In like manner for H, we substitute CH' where IF is the reading*

' " O

of the magnetometer on its arbitrary scale, and C is an unknown but constant multiplier which converts these readings into absolute measure, so that CH' is the horizontal force at a given instant.

The experiments to determine the absolute values of the quan tities must be conducted at a sufficient distance from the declino meter and magnetometer, so that the different magnets may not sensibly disturb each other. The time of every observation must be noted and the corresponding values of 8' and H' inserted. The equations are then to be treated so as to find 80, the constant error of the declinometer, and C the coefficient to be applied to the readings of the magnetometer. When these are found the readings of both instruments may be expressed in absolute measure. The absolute measurements, however, must be frequently repeated in order to take account of changes which may occur in the magnetic axis and magnetic moment of the magnets.

461.] The methods of determining the vertical component of the terrestrial magnetic force have not been brought to the same degree of precision. The vertical force must act on a magnet which turns about a horizontal axis. Now a body which turns about a hori zontal axis cannot be made so sensitive to the action of small forces as a body which is suspended by a fibre and turns about a vertical axis. Besides this, the weight of a magnet is so large compared

112 MAGNETIC MEASUREMENTS. [461.

with the magnetic force exerted upon it that a small displace ment of the centre of inertia by unequal dilatation, &c. produces a greater effect on the position of the magnet than a considerable change of the magnetic force.

Hence the measurement of the vertical force, or the comparison of the vertical and the horizontal forces, is the least perfect part of the system of magnetic measurements.

The vertical part of the magnetic force is generally deduced from the horizontal force by determining the direction of the total force.

If i be the angle which the total force makes with its horizontal component, i is called the magnetic Dip or Inclination, and if H is the horizontal force already found, then the vertical force is //tan i, and the total force is H sec i.

The magnetic dip is found by means of the Dip Needle.

The theoretical dip-needle is a magnet with an axis which passes through its centre of inertia perpendicular to the magnetic axis of the needle. The ends of this axis are made in the form of cylinders of small radius, the axes of which are coincident with the line passing through the centre of inertia. These cylindrical ends rest on two horizontal planes and are free to roll on them.

When the axis is placed magnetic east and west, the needle is free to rotate in the plane of the magnetic meridian, and if the instrument is in perfect adjustment, the magnetic axis will set itself in the direction of the total magnetic force.

It is, however, practically impossible to adjust a dip-needle so that its weight does not influence its position of equilibrium, because its centre of inertia, even if originally in the line joining the centres of the rolling sections of the cylindrical ends, will cease to be in this line when the needle is imperceptibly bent or un equally expanded. Besides, the determination of the true centre of inertia of a magnet is a very difficult operation, owing to the interference of the magnetic force with that of gravity.

Let us suppose one end of the needle and one end of the pivot to be marked. Let a line, real or imaginary, be drawn on the needle, which we shall call the Line of Collimation. The position of this line is read off on a vertical circle. Let 6 be the angle which this line makes with the radius to zero, which we shall suppose to be horizontal. Let A. be the angle which the magnetic axis makes with the line of collimation, so that when the needle is in this position the line of collimation is inclined 0 + A. to the horizontal.

461.] DIP CIRCLE. 11.3

Let p be the perpendicular from the centre of inertia on the plane on which the axis rolls, then p will be a function of 6, whatever be the shape of the rolling surfaces. If both the rolling sections of the ends of the axis are circular,

p — c — #sin(0+a) (1)

where a is the distance of the centre of inertia from the line joining the centres of the rolling sections, and a is the angle which this line makes with the line of collimation.

If M is the magnetic moment, m the mass of the magnet, and g the force of gravity, I the total magnetic force, and i the dip, then, by the conservation of energy, when there is stable equilibrium,

MIcos(0 + \ — i) — mgjp (2)

must be a maximum with respect to 0, or

MIsm(0 + -i)=-m<?d^> (3)

= —mg a cos (6 + a), if the ends of the axis are cylindrical.

Also, if T be the time of vibration about the position of equi librium,

: /,x

MI+ mga sin (6+ a) = -^-

where A is the moment of inertia of the needle about its axis of rotation.

In determining the dip a reading is taken with the dip circle in the magnetic meridian and with the graduation towards the west. Let 61 be this reading, then we have

MIsm(01 + \—i) = —m(/acos(0l + a). (5)

The instrument is now turned about a vertical axis through 180°, so that the graduation is to the east, and if 02 is the new reading, MIsm(02 + X — v+i) ——mga cos (02 + a). (6)

Taking (6) from (5), and remembering that 6^ is nearly equal to i, and 02 nearly equal to TT— i, and that X is a small angle, such that mgaK may be neglected in comparison with MI,

MI(0l—02-{-7f—2i') =—2mgaco$icosa. (7)

Now take the magnet from its bearings and place it in the deflexion apparatus, Art. 453, so as to indicate its own magnetic moment by the deflexion of a suspended magnet, then

M=\r*HD (8)

where D is the tangent of the deflexion.

VOL. II. I

114 MAGNETIC MEASUREMENTS. [461.

Next, reverse the magnetism of the needle and determine its new magnetic moment M', by observing a new deflexion, the tan gent of which is D' ', M> = i ^ H1)^ (9)

whence MD' = M'D. ( 1 0)

Then place it on its bearings and take two readings, 03 and 04, in which 03 is nearly ir + i, and 04 nearly —i,

3/'/' sin (03 + A' — 77 — i) = mgaco8(0B+a), (11)

M'l' sin (04 + A' + i) = m g a cos (04 + a), (1 2)

whence, as before,

M'I(93— 04 — 77 — 2i) — 2mgacosicosa, (13)

adding (8),

Jf/^-^ + Tr — 2z') + lT/(03 — 04— IT — 2 a) = 0, (14) or J9(01-02 + 7r-2;) + .Z/(03-04-7r-2*) = 0, (15)

whence we find the dip

-e4-Tr) , .

where D and _Z/ are the tangents of the deflexions produced by the needle in its first and second magnetizations respectively.

In taking observations with the dip circle the vertical axis is carefully adjusted so that the plane bearings upon which the axis of the magnet rests are horizontal in every azimuth. The magnet being magnetized so that the end A dips, is placed with its axis on the plane bearings, and observations are taken with the plane of the circle in the magnetic meridian, and with the graduated side of the circle east. Each end of the magnet is observed by means of reading microscopes carried on an arm which moves concentric with the dip circle. The cross wires of the microscope are made to coincide with the image of a mark on the magnet, and the position of the arm is then read off on the dip circle by means of a vernier.

We thus obtain an observation of the end A and another of the end B when the graduations are east. It is necessary to observe both ends in order to eliminate any error arising from the axle of the magnet not being concentric with the dip circle.

The graduated side is then turned west, and two more observ ations are made.

The magnet is then turned round so that the ends of the axle are reversed, and four more observations are made looking at the other side of the magnet.

463.] JOULE'S SUSPENSION. 115

The magnetization of the magnet is then reversed so that the end B dips, the magnetic moment is ascertained, and eight ohserva- tions are taken in this state, and the sixteen observations combined to determine the true dip.

462.] It is found that in spite of the utmost care the dip, as thus deduced from observations made with one dip circle, differs per ceptibly from that deduced from observations with another dip circle at the same place. Mr. Broun has pointed out the effect due to ellipticity of the bearings of the axle, arid how to correct it by taking observations with the magnet magnetized to different strengths.

The principle of this method may be stated thus. We shall suppose that the error of any one observation is a small quantity not exceeding a degree. We shall also suppose that some unknown but regular force acts upon the magnet, disturbing it from its true position.

If L is the moment of this force, 00 the true dip, and 0 the observed dip, then

L = Jf/sin(0-00), (17)

= MI(0-00), (18)

since 0 — 6$ is small.

It is evident that the greater M becomes the nearer does the needle approach its proper position. Now let the operation of taking the dip be performed twice, first with the magnetization equal to Mlt the greatest that the needle is capable of, and next with the magnetization equal to M~29 a much smaller value but sufficient to make the readings distinct and the error still moderate. Let 01 and 62 be the dips deduced from these two sets of observ ations, and let L be the mean value of the unknown disturbing force for the eight positions of each determination, which we shall suppose the same for both determinations. Then

L = M1i(01-e0) = M2i(02-00). (19)

If we find that several experiments give nearly equal values for L, then we may consider that 00 must be very nearly the true value of the dip.

463.] Dr. Joule has recently constructed a new dip-circle, in which the axis of the needle, instead of rolling on horizontal agate planes, is slung on two filaments of silk or spider's thread, the ends

I 2

116

MAGNETIC MEASUREMENTS.

[463-

of the filaments being attached to the arms of a delicate balance. The axis of the needle thus rolls on two loops of silk fibre, and Dr. Joule finds that its freedom of motion is much greater than when it rolls on agate planes.

In Fig. 18, NS is the needle, CC' is its axis, consisting of a straight cylindrical wire, and PCQ, P'C'Q' are the filaments on which

the axis rolls. POQ is the balance, consisting of a double bent lever supported by a wire, 0 0, stretched horizont ally between the prongs of a forked piece, and having a counterpoise It which can be screwed up or down, so that the balance is in neutral equilibrium about 0 0.

In order that the needle may be in neutral equilibrium as the needle rolls on the filaments the centre of gra vity must neither rise nor fall. Hence the distance OC must remain constant as the needle rolls. This condition will be fulfilled if the arms of the balance OP and 0 Q are equal, and if the filaments are at right angles to the arms.

Dr. Joule finds that the needle should not be more than five inches long. When it is eight inches long, the bending of the needle tends to diminish the apparent dip by a fraction of a minute. The axis of the needle was originally of steel wire, straightened by being brought to a red heat while stretched by a weight, but Dr. Joule found that with the new suspension it is not necessary to use steel wire, for platinum and even standard gold are hard enough.

The balance is attached to a wire 00 about a foot long stretched horizontally between the prongs of a fork. This fork is turned round in azimuth by means of a circle at the top of a tripod which supports the whole,. Six complete observations of the dip can be

464.]

VEETICAL FORCE.

117

obtained in one hour, and the average error of a single observation is a fraction of a minute of arc.

It is proposed that the dip needle in the Cambridge Physical Laboratory shall be observed by means of a double image instru ment, consisting of two totally reflecting prisms placed as in Fig. 19 and mounted on a vertical graduated circle, so that the plane of reflexion may be turned round a horizontal axis nearly coinciding with the prolongation of the axis of the suspended dip- needle. The needle is viewed by means of a telescope placed behind the prisms, and the two ends of the needle are seen together as in Fig. 20. By turning the prisms about the axis of the vertical circle, the images of two lines drawn on the needle may be made to coincide. The inclination of the needle is thus determined from the reading of the vertical circle.

Fig. 19.

Fig. 20.

The total intensity / of the magnetic force in the line of dip may be deduced as follows from the times of vibration in the four positions already described,

T13 Tz, jP3,

5JL _L JL J_l.

" 2M+2' I Zi2 + Tf "h T* h T* )

The values of M and M' must be found by the method of deflexion and vibration formerly described, and A is the moment of inertia of the magnet about its axle.

The observations with a magnet suspended by a fibre are so much more accurate that it is usual to deduce the total force from the horizontal force from the equation

/= H sec 6,

where / is the total force, H the horizontal force, and 0 the dip.

464.] The process of determining the dip being a tedious one, is not suitable for determining the continuous variation of the magnetic

118 MAGNETIC MEASUREMENTS. [464.

force. The most convenient instrument for continuous observa tions 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 0 the small angle which the magnetic axis makes with the horizon

HZ = mgacQ&(a.~6),

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 bZ, there will be a variation of the angular position of the magnet bO such that

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

when 0 = 0, and the value of -y-r •

civ

The value of Z, when the horizontal force and the dip are known, is found from the equation Z = ZTtan00, where 00 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 i\ east or west from the declinometer, as in ex periments on deflexion, and let the tangent of deflexion be Dl .

Then place it with its axis vertical and with its centre at a distance rz above or below the centre of the vertical force mag netometer, and let the tangent of the deflexion produced in the magnetometer be D2. Then, if the moment of the deflecting

magnet is M, jr.

M^IIr^D^ = ^r^D2.

clZ r^ DL

Hence -7— = H -^ -~ •

dO r23 D2

The actual value of the vertical force at any instant is 7 7 +fidZ

& = &Q H- v -j^ >

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

For continuous observations of the variations of magnetic force

464.] VERTICAL FOKCE. 119

at a fixed observatory the Unifilar Declinometer, the Bifilar Hori zontal Force Magnetometer, and the Balance Vertical Force Mag netometer 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 com ponents of the force from their standard values. The declinometer gives the force towards mean magnetic west, the bifilar magnet ometer 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 sur face 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, X=Hcosb, directed due north, \

Y=Hsmb, directed due west, (1)

Z = If tan 0, directed vertically downwards, ) where H denotes the horizontal force, 8 the declination, and 0 the dip.

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

Y i dr i dv dv , }

A = -- ^yj Y = - j — > ^=-^-7 (*)

a dl a cos I dK dr

where I is the latitude, and A. the longitude, and r 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 FQ be the value of V at the true north pole, then, taking the line-integral along any meridian, we find,

o, (3)

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

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 F0, the value of Fat 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 F0.

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 T over the whole surface.

Let JF»»/j:tf+7*, W

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

F= ^+«fVco8/#A, (5)

^AO

where the integration is performed along the parallel I 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 com ponents 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 con siderable number of stations distributed fairly over the country.

Within this district we may suppose the value of V to be re presented with sufficient accuracy by the formula

F= Vt + a(AJ + Ai+\BJ*+EJ+\3iK* + ^ (6)

whence X = A1 + Bl I + £2 X, (7)

Ycosl = A2 + £2l + 33. (8)

Let there be n stations whose latitudes are ll} £2, ...&c. and longitudes \lt A2, &c., and let X and 7 be found for each station.

Let J =

122 TERRESTRIAL MAGNETISM. [466-

/0 and A0 may be called the latitude and longitude of the central station. Let

X0=-i(i)- and rocosJ0=:-2(rcosJ), (10)

tl ti

then X0 and Y0 are the values of X and Y at the imaginary central station, then

-\0), (11)

A-A0). (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 £, and that of Ycos I by q, then we may calculate f and r/ on the supposition that they arise from errors of observation of H and 8.

Let the probable error of H be ^, and that of 8, d, then since

dX — cos 5 . dff—Hsm 8 . db, £2 = 7,2 COS2 8 + dH sin2 8

Similarly 7?2 = /fc2 sin2 8 + d2 //2 cos2 8.

If the variations of X and T from their values as given by equa tions 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 £ to r\ any other value than unity.

According to the method of least squares we multiply the equa tions of the form (11) by r/, and those of the form (12) by £ to make their probable error the same. We then multiply each equation by the coefficient of one of the unknown quantities J3lt H2, or BZ and add the results, thus obtaining three equations from which to find B B and B.

in which we write for conciseness,

*1 = 2(^2)-»^ ^ = Pl = 2(lX)-nlQXQ,

By calculating £19 J52, and J53, 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

468.] MAGNETIC FEATURES OF THE EARTH. 123

which are found to exist where the rock near the station is magnetic, as most igneous rocks are.

Surveys of this kind can be made only in countries where mag netic instruments can be carried about and set up in a great many stations. For other parts of the world we must be content to find the distribution of the magnetic elements by interpolation between their values at a few stations at great distances from each other.