If we adopt the first of these two hypotheses, which will be nearly true if the wire itself nearly fills up the whole space, then we may put y = ax, Y= $y,

where a and ft are constant numerical quantities, and

where a is a constant depending upon the size and form of the free space left inside the coil.

Hence, if we make the thickness of the wire vary in the same ratio as as, we obtain very little advantage by increasing the external size of the coil after the external dimensions have become a large multiple of the internal dimensions.

720.] If increase of resistance is not regarded as a defect, as when the external resistance is far greater than that of the gal vanometer, or when our only object is to produce a field of intense force, we may make y and Y constant. We have then

N

G= 71 (*-")>

-p 1 Pf/*.3 n %\

~ 3 Yf> Jj'* * ''

where a is a constant depending on the vacant space inside the coil. In this case the value of G increases uniformly as the dimensions of the coil are increased, so that there is no limit to the value of G except the labour and expense of making the coil.

326

ELECTROMAGNETIC INSTRUMENTS,

[721.

On Suspended Coils.

721.] In the ordinary galvanometer a suspended magnet is acted on by a fixed coil. But if the coil can be suspended with sufficient delicacy, we may determine the action of the magnet, or of another coil on the suspended coil, by its deflexion from the position of equilibrium.

We cannot, however, introduce the electric current into the coil unless there is metallic connexion between the electrodes of the battery and those of the wire of the coil. This connexion may be made in two different ways, by the Bifilar Suspension, and by wires in opposite directions.

The bifilar suspension has already been described in Art. 459 as applied to magnets. The arrangement of the upper part of the suspension is shewn in Fig. 55. When applied to coils, the two fibres are no longer of silk but of metal, and since the torsion of a metal wire capable of supporting the coil and transmitting the current is much greater than that of a silk fibre, it must be taken specially into account. This suspension has been brought to great perfection in the instruments constructed by M. Weber.

The other method of suspension is by means of a single wire which is connected to one extremity of the coil. The other ex tremity of the coil is connected to another wire which is made to hang down, in the same vertical straight line with the first wire, into a cup of mercury, as is shewn in Fig. 57, Art. 729. In certain cases it is convenient to fasten the extremities of the two wires to pieces by which they may be tightly stretched, care being taken

that the line of these wires passes through the centre of gravity of the coil. The apparatus in this form may be used when the axis is not vertical ; see Fig. 53.

722.] The suspended coil may be used as an exceedingly sensitive gal vanometer, for, by increasing the in tensity of the magnetic force in the field in which it hangs, the force due to a feeble current in the coil may be greatly increased without adding to the mass of the coil. The mag netic force for this purpose may be

Fig. 53.

produced by means of permanent magnets, or by electromagnets

723-] SUSPENDED COIL. 327

excited by an auxiliary current, and it may be powerfully concen trated on the suspended coil by means of soft iron armatures. Thus, in Sir W. Thomson's recording apparatus, Fig. 53, the coil is sus pended between the opposite poles of the electromagnets N and S, and in order to concentrate the lines of magnetic force on the ver tical sides of the coil, a piece of soft iron, 1), is fixed between the poles of the magnets. This iron becoming magnetized by induc tion, produces a very powerful field of force, in the intervals between it and the two magnets, through which the vertical sides of the coil are free to move, so that the coil, even when the current through it is very feeble, is acted on by a considerable force tending to turn it about its vertical axis.

723.] Another application of the suspended coil is to determine, by comparison with a tangent galvanometer, the horizontal com ponent of terrestrial magnetism.

The coil is suspended so that it is in stable equilibrium when its plane is parallel to the magnetic meridian. A current y is passed through the coil and causes it to be deflected into a new position of equilibrium, making an angle 0 with the magnetic meridian. If the suspension is bifilar, the moment of the couple which produces this deflexion is I1 sin 0, and this must be equal to HyffcosO, where His the horizontal component of terrestrial mag netism, y is the current in the coil, and g is the sum of the areas of all the windings of the coil. Hence

F

II y — — tan0.

g

If A is the moment of inertia of the coil about its axis of sus pension, and Tthe time of a single vibration, FT2 = v*A,

Ti^A

and we obtain Hy = - tan 0.

If the same current passes through the coil of a tangent galva nometer, and deflects the magnet through an angle 0,

y

where G is the principal constant of the tangent galvanometer, Art. 710, From these two equations we obtain 7T tr /AGkaxid TT /A tan 0 tan rf>

: ~T A/Tte^T' = T V -oT

Tliis method wa^ given by F. Kohlrausch *.

• r"ogg., Ann. cxxxviii, Feb. 18G9.

328 ELECTROMAGNETIC INSTRUMENTS. [?24-

724.] Sir William Thomson has constructed a single instrument by means of which the observations required to determine H and y may be made simultaneously by the same observer.

The coil is suspended so as to be in equilibrium with its plane in the magnetic meridian, and is deflected from this position when the current flows through it. A very small magnet is sus pended at the centre of the coil, and is deflected by the current in the direction opposite to that of the deflexion of the coil. Let the deflexion of the coil be 6, and that of the magnet 0, then the energy of the system is

Hy g sm9 + my G sin (0 — fy — Hmcos 0 — Fcos 9.

Differentiating with respect to 0 and 0, we obtain the equa tions of equilibrium of the coil and of the magnet respectively,

Hyg cos 0 + my (7 cos (0 — 0) + F sin Q = 0, — my G cos (6 — 0)-f Hm sin 0 = 0.

From these equations we find, by eliminating H or y} a quadratic equation from which y or // may be found. If m, the magnetic moment of the suspended mag-net, is very small, we obtain the following approximate values

j _ IT /— ^<?sin0cos(0 — 0) L mG cos (6 — 0)

' ~T 'V g cos 6 sin 0 2 g cos0

77 / — ^4 sin 0 sin 0 ^m sin0

" ~T V G g cos 6 cos (0—0) ~~ 2 7 cos^ "

In these expressions G and g are the principal electric constants of the coil, A its moment of inertia, T its time of vibration, m the magnetic moment of the magnet, H the intensity of the horizontal magnetic force, y the strength of the current, 0 the deflexion of the coil, and 0 that of the magnet.

Since the deflexion of the coil is in the opposite direction to the deflexion of the magnet, these values of H and y will always be real.

Weber's Electrody nanometer.

725.] In this instrument a small coil is suspended by two wires within a larger coil which is fixed. When a current is made to flow through both coils, the suspended coil tends to place itself parallel to the fixed coil. This tendency is counteracted by the moment of the forces arising from the bifilar suspension, and it is also affected by the action of terrestrial magnetism on the sus pended coil.

725.] ELECTRODYNAMOMETER. 329

In the ordinary use of the instrument the planes of the two coils are nearly at right angles to each other, so that the mutual action of the currents in the coils may be as great as possible, and the plane of the suspended coil is nearly at right angles to the magnetic meridian, so that the action of terrestrial magnetism may be as small as possible.

Let the magnetic azimuth of the plane of the fixed coil be a, and let the angle which the axis of the suspended coil makes with the plane of the fixed coil be Q + fi, where (3 is the value of this angle when the coil is in equilibrium and no current is flowing, and"* 6 is the deflexion due to the current. The equation of equi librium is

Let us suppose that the instrument is adjusted so that a and j3 are both very small, and that Hgy^ is small compared with F. We have in this case, approximately,

(r^y1y2cos/3 Zfyy2sin(a-|-/3) HGg^y^y^ G2y2yl2y22smj3

If the deflexions when the signs of yl and y2 are changed are as follows : e when is , and ,

then we find

F

yl y2 — J — — (tan 0J + tan 02 — tan 03 — tan 04).

If it is the same current which flows through both coils we may put yl y2 = y2, and thus obtain the value of y.

When the currents are not very constant it is best to adopt this method, which is called the Method of Tangents.

If the currents are so constant that we can adjust /3, the angle of the torsion-head of the instrument, we may get rid of the correction for terrestrial magnetism at once by the method of sines. In this method /3 is adjusted till the deflexion is zero, so that

0=_/3.

If the signs of y1 and y2 are indicated by the suffixes of /3 as before,

Fsin & = -Fsin P3 = — Gffyly2 + Hg y2 sin a,

F sin )32 = — ^sin /34 = — Gg yl y^ — Rg y2 sin a,

F

and Yl y2 = - — ^ (sin fa + sin fa - sin fa - sin fa).

330

ELECTROMAGNETIC INSTRUMENTS.

[725<

725.] ELECTRODYNAMOMETER. 331

This is the method adopted by Mr. Latimer Clark in his use of the instrument constructed by the Electrical Committee of the British Association. We are indebted to Mr. Clark for the drawing of the electrodynamometer in Figure 54, in which Helmholtz's arrangement of two coils is adopted both for the fixed and for the suspended coil*. The torsion-head of the instrument, by which the bifilar suspension is adjusted, is represented in Fig. 55. The

Fig. 55.

equality of the tension of the suspension wires is ensured by their being attached to the extremities of a silk thread which passes over a wheel, and their distance is regulated by two guide-wheels, which can be set at the proper distance. The suspended coil can be moved vertically by means of a screw acting on the suspension-wheel, and horizontally in two directions by the sliding pieces shewn at the bottom of Fig. 55. It is adjusted in azimuth by means of the torsion-screw, which turns the torsion-head round a vertical axis (see Art. 459). The azimuth of the suspended coil is ascertained by observing the reflexion of a scale in the mirror, shewn just beneath the axis of the suspended coil.

• In the actual instrument, the wires conveying the current to and from the coils are not spread out as displayed in the figure, but are kept as close together as pos sible, so as to neutralize each other's electromagnetic action.

332 ELECTROMAGNETIC INSTRUMENTS.

The instrument originally constructed by Weber is described in his Elektroctynamiscke Maasbeslimmungen. It was intended for the measurement of small currents, and therefore both the fixed and the suspended coils consisted of many windings, and the suspended coil occupied a larger part of the space within the fixed coil than in the instrument of the British Association, which was primarily in tended as a standard instrument, with which more sensitive instru ments might be compared. The experiments which he made with it furnish the most complete experimental proof of the accuracy of Ampere's formula as applied to closed currents, and form an im portant part of the researches by which Weber has raised the numerical determination of electrical quantities to a very high rank as regards precision.

Weber's form of the electrodynarnometer, in which one coil is suspended within another, and is acted on by a couple tending to turn it about a vertical axis, is probably the best fitted for absolute measurements. A method of calculating the constants of such an arrangement is given in Art. 697.

726.] If, however, we wish, by means of a feeble current, to produce a considerable electromagnetic force, it is better to place the suspended coil parallel to the fixed coil, and to make it capable of motion to or from it.

The suspended coil in Dr. Joule's current- weigher, Fig. 56, is horizontal, and capable of vertical motion, and the force between it and the fixed coil is estimated by the weight which must be added to or removed from the coil in order to bring it to the same relative position with respect to the fixed coil that it has when no current passes.

The suspended coil may also be fastened to the extremity of the hori- 56< zontal arm of a torsion-balance, and

may be placed between two fixed coils, one of which attracts it, while the other repels it, as in Fig. 57.

By arranging the coils as described in Art. 729, the force acting on the suspended coil may be made nearly uniform within a small distance of the position of equilibrium.

Another coil may be fixed to the other extremity of the arm of the torsion-balance and placed between two fixed coils. If the

728.]

CURRENT-WEIGHER.

333

two suspended coils are similar, but with the current flowing in opposite directions, the effect of terrestrial magnetism on the

Fig. 57.

position of the arm of the torsion-balance will be completely eliminated.

727.] If the suspended coil is in the shape of a long solenoid, and is capable of moving parallel to its axis, so as to pass into the interior of a larger fixed solenoid having the same axis, then, if the current is in the same direction in both solenoids, the sus pended solenoid will be sucked into the fixed one by a force which will be nearly uniform as long as none of the extremities of the solenoids are near one another.

728.] To produce a uniform longitudinal force on a small coil placed between two equal coils of much larger dimensions, we should make the ratio of the diameter of the large coils to the dis tance between their planes that of 2 to /3. If we send the same current through these coils in opposite directions, then, in the ex pression for o>, the terms involving odd powers of r disappear, and since sin2 a = -f and cos2 a = f, the term involving /-4 disappears also, and we have

~ Q2 (0) + V

&c

which indicates a nearly uniform force on a small suspended coil. The arrangement of the coils in this case is that of the two outer coils in the galvanometer with three coils, described at Art. 715. See Fig. 51.

334 ELECTROMAGNETIC INSTRUMENTS. [?29-

729.] If we wish to suspend a coil between two coils placed so near it that the distance between the mutually acting wires is small compared with the radius of the coils, the most uniform force is obtained by making the radius of either of the outer coils exceed

that of the middle one by — - ^ of the distance between the planes

v3 of the middle and outer coils.

CHAPTER XVI.

ELECTROMAGNETIC OBSERVATIONS.

730.] So many of the measurements of electrical quantities depend on observations of the motion of a vibrating body that we shall devote some attention to the nature of this motion, and the best methods of observing it.

The small oscillations of a body about a position of stable equi librium are, in general, similar to those of a point acted on by a force varying directly as the distance from a fixed point. In the case of the vibrating bodies in our experiments there is also a resistance to the motion, depending on a variety of causes, such as the viscosity of the air, and that of the suspension fibre. In many electrical instruments there is another cause of resistance, namely, the reflex action of currents induced in conducting circuits placed near vibrating magnets. These currents are induced by the motion of the magnet, and their action on the magnet is, by the law of Lenz, invariably opposed to its motion. This is in many cases the principal part of the resistance.

A metallic circuit, called a Damper, is sometimes placed near a magnet for the express purpose of damping or deadening its vibrations. We shall therefore speak of this kind of resistance as Damping.

In the case of slow vibrations, such as can be easily observed, the whole resistance, from whatever causes it may arise, appears to be proportional to the velocity. It is only when the velocity is much greater than in the ordinary vibrations of electromagnetic instruments that we have evidence of a resistance proportional to the square of the velocity.

We have therefore to investigate the motion of a body subject to an attraction varying as the distance, and to a resistance varying as the velocity.

336

ELECTROMAGNETIC OBSERVATIONS.

731.] The following application, by Professor Tait*, of the principle of the Hodograph, enables us to investigate this kind of motion in a very simple manner by means of the equiangular spiral.

Let it be required to find the acceleration of a particle which describes a logarithmic or equiangular spiral with uniform angular velocity o> about the pole.

The property of this spiral is, that the tangent PT makes with the radius vector PS a constant angle a.

If v is the velocity at the point P, then

v . sin a = co . SP.

Hence, if we draw SP' parallel to PT and equal to SP, the velocity at P will be given both in magnitude and direction by

v =

sin a

•SP.

Fig. 58.

Hence P' will be a point in the hodograph. But SP is SP turned through a constant angle TT — a, so that the hodograph described by P is the same as the original spiral turned about its pole through an angle TT — a.

The acceleration of P is represented in magnitude and direction

by the velocity of P' multiplied by the same factor, -.

Hence, if we perform on SP the same operation of turning it

• Proc. R. S. Win., Dec. 16, 1867.

732.] DAMPED VIBRATIONS. 337

through an angle IT — a into the position SP', the acceleration of P will be equal in magnitude and direction to

-£•'&,

where SP' is equal to SP turned through an angle 2 IT — 2 a.

If we draw PF equal and parallel to SP', the acceleration will be

9

PF, which we may resolve into

sin2 a

J?LpSn& -4-P. sm*a sin^a

The first of these components is a central force towards S pro portional to the distance.

The second is in a direction opposite to the velocity, and since

_, sin a cos a PK = 2 cos a PS = - 2 - v,

0}

this force may be written

co cos a

— 2—. v.

sin a

The acceleration of the particle is therefore compounded of two parts, the first of which is an attractive force /ur, directed towards S, and proportional to the distance, and the second is — 2 kv, a resist ance to the motion proportional to the velocity, where

ft)2 , 7 cos a

a = . , and k = o> -.

sin^ a sin a

If in these expressions we make a = — , the orbit becomes a circle,

and we have JUG = o)02, and k = 0.

Hence, if the law of attraction remains the same, ju = /ut0 , and

co = o)0 sin a,

or the angular velocity in different spirals with the same law of attraction is proportional to the sine of the angle of the spiral.

732.] If we now consider the motion of a point which is the projection of the moving point P on the horizontal line XT, we shall find that its distance from S and its velocity are the horizontal components of those of P. Hence the acceleration of this point is also an attraction towards S, equal to /x, times its distance from Sf together with a retardation equal to k times its velocity.

We have therefore a complete construction for the rectilinear motion of a point, subject to an attraction proportional to the distance from a fixed point, and to a resistance proportional to the velocity. The motion of such a point is simply the horizontal

VOL. II. Z

338 ELECTROMAGNETIC OBSERVATIONS. [733.

part of the motion of another point which moves with uniform angular velocity in a logarithmic spiral.

733.] The equation of the spiral is

r = Ce-$CQia.

To determine the horizontal motion, we put <£ = co ^, x = a--r sin </>, where a is the value of x for the point of equilibrium.

If we draw BSD making an angle a with the vertical, then the tangents BX> DY, GZ, &c. will be vertical, and X, Y, Z, &c. will be the extremities of successive oscillations.

734.] The observations which are made on vibrating bodies are —

(1) The scale-reading at the stationary points. These are called

Elongations.

(2) The time of passing a definite division of the scale in the

positive or negative direction.

(3) The scale-reading at certain definite times. Observations of

this kind are not often made except in the case of vibrations of long period *. The quantities which we have to determine are —

(1) The scale-reading at the position of equilibrium.

(2) The logarithmic decrement of the vibrations.

(3) The time of vibration.

To determine the Reading at the Position of Equilibrium from Three Consecutive Elongations,

735.] Let #!, #2, #3 be the observed scale-readings, corresponding to the elongations X, Y, Z, and let a be the reading at the position of equilibrium, S, and let r^ be the value of SB,

j — a = /! sin a,

$2 — a = — 1\ sin a e~* cot a, #3 — a = rl sina£-27rcota. From these values we find

(!-«) 08 -«) = 0*2-«)2»

, X-,

whence a = —

vU\ "J~ «2/o "™ * .— &O

When a*3 does not differ much from x^ we may use as an ap proximate formula

a = }(a?1 + 2a?a + a?3).

• See Gauss, Resultate des Magnetischen Vereins, 1836. II.

LOGAKITHMIC DECREMENT. 339

To determine the Logarithmic Decrement.

736.] The logarithm of the ratio of the amplitude of a vibration to that of the next following is called the Logarithmic Decrement. If we write p for this ratio

L is called the common logarithmic decrement, and A. the Napierian logarithmic decrement. It is manifest that A = L loge 10 = 77 cot a.

Hence a = cot"1-

77

which determines the angle of the logarithmic spiral.

In making a special determination of A we allow the body to perform a considerable number of vibrations. If c1 is the amplitude of the first, and cn that of the n^ vibration,

If we suppose the accuracy of observation to be the same for small vibrations as for large ones, then, to obtain the best value of A, we should allow the vibrations to subside till the ratio of c1 to cn becomes most nearly equal to e, the base of the Napierian

logarithms. This gives n the nearest whole number to - + 1 .

A

Since, however, in most cases time is valuable, it is best to take the second set of observations before the diminution of amplitude has proceeded so far.

737.] In certain cases we may have to determine the position of equilibrium from two consecutive elongations, the logarithmic decrement being known from a special experiment. We have then

_ #l + £ ^2

Time of Vibration .

738.] Having determined the scale-reading of the point of equi librium, a conspicuous mark is placed at that point of the scale, or as near it as possible, and the times of the passage of this mark are noted for several successive vibrations.

Let us suppose that the mark is at an unknown but very small distance as on the positive side of the point of equilibrium, and that

z 2

340 ELECTROMAGNETIC OBSERVATIONS. [739.

tfj is the observed time of the first transit of the mark in the positive direction, and £2, ^3, &c. the times of the following transits.

If T be the time of vibration, and P15 P2, P3, &c. the times of transit of the true point of equilibrium,

where vlt v29 &c. are the successive velocities of transit, which we may suppose uniform for the very small distance SB.

If p is the ratio of the amplitude of a vibration to the next in

succession, 1 , as x

v9 — -- #T , and. — = — p — •

P l /2 ^l

If three transits are observed at times ti3 t2, £3, we find

The period of vibration is therefore

2P+1 The time of the second passage of the true point of equilibrium is

P2 = i (^-f 2 ^2 + O ~i / " \z (*i — 2 ^2 + ^)-

Three transits are sufficient to determine these three quantities, but any greater number may be combined by the method of least squares. Thus, for five transits,

The time of the third transit is,

739.] The same method may be extended to a series of any number of vibrations. If the vibrations are so rapid that the time of every transit cannot be recorded, we may record the time of every third or every fifth transit, taking care that the directions of successive transits are opposite. If the vibrations continue regular for a long time, we need not observe during the whole time. We may begin by observing a sufficient number of transits to determine approximately the period of vibration, T, and the time of the middle transit, P, noting whether this transit is in the positive or the negative direction. We may then either go on counting the vibrations without recording the times of transit, or we may leave the apparatus un watched. We then observe a

PERIODIC TIME OF VIBRATION. 341

second series of transits,, and deduce the time of vibration T' and the time of middle transit P', noting the direction of this transit.

If T and Tf, the periods of vibration as deduced from the two sets of observations, are nearly equal, we may proceed to a more accurate determination of the period by combining the two series of observations.

Dividing P'— P by T, the quotient ought to be very nearly an integer, even or odd according as the transits P and P' are in the same or in opposite directions. If this is not the case, the series of observations is worthless, but if the result is very nearly a whole number n, we divide P'— P by n, and thus find the mean value of T for the whole time of swinging.

740.] The time of vibration T thus found is the actual mean time of vibration, and is subject to corrections if we wish to deduce from it the time of vibration in infinitely small arcs and without damping.

To reduce the observed time to the time in infinitely small arcs, we observe that the time of a vibration of amplitude a is in general of the form T - T^(l + *c2),

where K is a coefficient, which, in the case of the ordinary pendulum, is -g^. Now the amplitudes of the successive vibrations are c, cp1f cp2, ... cpl~n, so that the whole time of n vibrations is

where T is the time deduced from the observations.

Hence, to find the time T^ in infinitely small arcs, we have approximately,

n p-! To find the time T0 when there is no damping, we have

sn a

741.] The equation of the rectilinear motion of a body, attracted to a fixed point and resisted by a force varying as the velocity, is

7 n j

.^ + 2*^+»(-«)=sO, (1)

where x is the coordinate of the body at the time t, and a is the coordinate of the point of equilibrium.

342 ELECTROMAGNETIC OBSERVATIONS. [?42-

To solve this equation, let

x-a = e-Vy; (2)

then gl + ^.^^o; (3)

the solution of which is

y — Ccos (/oo2— IP t--d), when k is less than <o ; (4)

y = A + Bt, when k is equal to o> ; (5)

and y — C' cos h ( Vk* — o>2 1 + a), when k is greater than o>. (6)

The value of a? may be obtained from that of y by equation (2). When k is less than o>, the motion consists of an infinite series of oscillations, of constant periodic time, but of continually decreasing amplitude. As k increases, the periodic time becomes longer, and the diminution of amplitude becomes more rapid.

When k (half the coefficient of resistance) becomes equal to or greater than o>, (the square root of the acceleration at unit distance from the point of equilibrium,) the motion ceases to be oscillatory, and during the whole motion the body can only once pass through the point of equilibrium, after which it reaches a position of greatest elongation, and then returns towards the point of equilibrium, con tinually approaching, but never reaching it.

Galvanometers in which the resistance is so great that the motion is of this kind are called dead beat galvanometers. They are useful in many experiments, but especially in telegraphic signalling, in which the existence of free vibrations would quite disguise the movements which are meant to be observed.

Whatever be the values of k and o>, the value of a, the scale- reading at the point of equilibrium, may be deduced from five scale- readings, p, q, r, s, t, taken at equal intervals of time, by the formula

(p-2+r) (r- 2s + 1) - (q-

On the Observation of the Galvanometer.

742.] To measure a constant current with the tangent galvano meter, the instrument is adjusted with the plane of its coils parallel to the magnetic meridian, and the zero reading is taken. The current is then made to pass through the coils, and the deflexion of the magnet corresponding to its new position of equilibrium is observed. Let this be denoted by $.

Then, if // is the horizontal magnetic force, G the coefficient of the galvanometer, and y the strength of the current,

(I)

744-] DEFLEXION OF THE GALVANOMETER. 343

If the coefficient of torsion of the suspension fibre is r MH (see Art. 452), we must use the corrected formula

JT

y = -(tan$+r(j[>sec<£). (2)

Best Value of the Deflexion.

743.] In some galvanometers the number of windings of the coil through which the current flows can be altered at pleasure. In others a known fraction of the current can be diverted from the galvanometer by a conductor called a Shunt. In either case the value of G, the effect of a unit-current on the magnet, is made to vary.

Let us determine the value of £, for which a given error in the observation of the deflexion corresponds to the smallest error of the deduced value of the strength of the current.

Differentiating equation (1), we find

dy H , .

4=^sec*-

Eliminating G, -~ = — sin 2 $. (4)

This is a maximum for a given value of y when the deflexion is 45°. The value of G should therefore be adjusted till Gy is as nearly equal to H as is possible ; so that for strong currents it is better not to use too sensitive a galvanometer.

On the Best Method of applying the Current.

744.] When the observer is able, by means of a key, to make or break the connexions of the circuit at any instant, it is advisable to operate with the key in such a way as to make the magnet arrive at its position of equilibrium with the least possible velocity. The following method was devised by Gauss for this purpose.

Suppose that the magnet is in its position of equilibrium, and that there is no current. The observer now makes contact for a short time, so that the magnet is set in motion towards its new position of equilibrium. He then breaks contact. The force is now towards the original position of equilibrium, and the motion is retarded. If this is so managed that the magnet comes to rest exactly at the new position of equilibrium,, and if the observer again makes con tact at that instant and maintains the contact, the magnet will remain at rest in its new position.

344 ELECTROMAGNETIC OBSERVATIONS. [745.

If we neglect the effect of the resistances and also the inequality of the total force acting in the new and the old positions, then, since we wish the new force to generate as much kinetic energy during the time of its first action as the original force destroys while the circuit is broken, we must prolong the first action of the current till the magnet has moved over half the distance from the first position to the second. Then if the original force acts while the magnet moves over the other half of its course, it will exactly stop it. Now the time required to pass from a point of greatest elongation to a point half way to the position of equilibrium is one-sixth of a complete period, or one-third of a single vibration.

The operator, therefore, having previously ascertained the time of a single vibration, makes contact for one-third of that time, breaks contact for another third of the same time, and then makes contact again during the continuance of the experiment. The magnet is then either at rest, or its vibrations are so small that observations may be taken at once, without waiting for the motion to die away. For this purpose a metronome may be adjusted so as to beat three times for each single vibration of the magnet.

The rule is somewhat more complicated when the resistance is of sufficient magnitude to be taken into account, but in this case the vibrations die away so fast that it is unnecessary to apply any corrections to the rule.

When the magnet is to be restored to its original position, the circuit is broken for one-third of a vibration, made again for an equal time, and finally broken. This leaves the magnet at rest in its former position.

If the reversed reading is to be taken immediately after the direct one, the circuit is broken for the time of a single vibration and then reversed. This brings the magnet to rest in the reversed position.

Measurement l>y the First Swing.

745.] When there is no time to make more than one observation, the current may be measured by the extreme elongation observed in the first swing of the magnet. If there is no resistance, the permanent deflexion $ is half the extreme elongation. If the re sistance is such that the ratio of one vibration to the next is p, and if 00 is the zero reading, and dl the extreme elongation in the first swing, the deflexion, <£, corresponding to the point of equilibrium is

0Q+P0!

9 1+p

747-] SERIES OF OBSERVATION'S. 345

In this way the deflexion may be calculated without waiting for the magnet to come to rest in its position of equilibrium.

To make a Series of Observations.