A Treatise on Electricity and Magnetism, Vol. 2 (1873) — part 21 of 27

746.] The best way of making a considerable number of mea sures of a constant current is by observing three elongations while the current is in the positive direction, then breaking contact for about the time of a single vibration, so as to let the magnet swing into the position of negative deflexion, then reversing the current and observing three successive elongations on the negative side, then breaking contact for the time of a single vibration and re peating the observations on the positive side, and so on till a suffi cient number of observations have been obtained. In this way the errors which may arise from a change in the direction of the earth's magnetic force during the time of observation are eliminated. The operator, by carefully timing the making and breaking of contact, can easily regulate the extent of the vibrations, so as to make them sufficiently small without being indistinct. The motion of the magnet is graphically represented in Fig. 59, where the abscissa represents the time, and the ordinate the deflexion of the magnet. If 01 . . . 06 be the observed elongations, the deflexion is given by the equation 8 = + 2 0 + 0_0_20 — 0.

Fig. 59.

Method of Multiplication.

747.] In certain cases, in which the deflexion of the galvanometer magnet is very small, it may be advisable to increase the visible effect by reversing the current at proper intervals, so as to set up a swinging motion of the magnet. For this purpose, after ascertaining the time, T, of a single vibration of the magnet, the current is sent in the positive direction for a time T, then in the reversed direction for an equal time, and so on. When the motion of the magnet has become visible, we may make the reversal of the current at the observed times of greatest elongation.

Let the magnet be at the positive elongation 00, and let the current be sent through the coil in the negative direction. The

346 ELECTROMAGNETIC OBSERVATIONS. [748.

point of equilibrium is then — $, and the magnet will swing to a negative elongation 0, such that

Similarly, if the current is now made positive while the magnet swings to 02, P02 = -01 + (p+ 1) 0,

or P202 = 00 + (P+1)24>; and if the current is reversed n times in succession, we find

whence we may find <£ in the form

**«FTf=7-

If ^ is a number so great that p~n may be neglected, the ex pression becomes n — 1

The application of this method to exact measurement requires an accurate knowledge of p, the ratio of one vibration of the magnet to the next under the influence of the resistances which it expe riences. The uncertainties arising from the difficulty of avoiding irregularities in the value of p generally outweigh the advantages of the large angular elongation. It is only where we wish to establish the existence of a very small current by causing it to produce a visible movement of the needle that this method is really valuable.

On the Measurement of Transient Currents.

748.] When a current lasts only during a very small fraction of the time of vibration of the galvanometer-magnet, the whole quan tity of electricity transmitted by the current may be measured by the angular velocity communicated to the magnet during the passage of the current, and this may be determined from the elongation of the first vibration of the magnet.

If we neglect the resistance which damps the vibrations of the magnet, the investigation becomes very simple.

Let y be the intensity of the current at any instant, and Q the quantity of electricity which it transmits, then

= \ydt. (1)

749-] TRANSIENT CURRENTS. 347

Let M be the magnetic moment, and A the moment of inertia of the magnet and suspended apparatus,

,72/9

A "L^ + MHsm 0 = MGy cos 0. (2)

(It

If the time of the passage of the current is very small, we may integrate with respect to t during this short time without regarding the change of 0, and we find

=MG cos 00 ydt + C = MGQ cos 00 + C. (3)

This shews that the passage of the quantity Q produces an angular momentum MGQ cos 00 in the magnet, where 00 is the value of 0 at the instant of passage of the current. If the magnet is initially in equilibrium, we may make 00 = 0.

The magnet then swings freely and reaches an elongation 01. If there is no resistance, the work done against the magnetic force during this swing is MR (I — cosflj.

The energy communicated to the magnet by the current is

Equating these quantities, we find

lf = 2^(l-cos<y, (4)

s-IJ- a ^ •*• » '

tf6- -^t

dO /MH .

whence -=- = 2 A / — — - sin J 0j

^ \ A

i/rn

Qby(3). (5)

A

But if T be the time of a single vibration of the magnet,

T

= " A/

(6)

TT m

and we find Q = — - - 2 sin \ Qlt (7)

where // is the horizontal magnetic force, Q- the coefficient of the galvanometer, T the time of a single vibration, and Ol the first- elongation of the magnet.

749.] In many actual experiments the elongation is a small angle, and it is then easy to take into account the effect of resist ance, for we may treat the equation of motion as a linear equation.

Let the magnet be at rest at its position of equilibrium, let an angular velocity v be communicated to it instantaneously, and let its first elongation be Ol .

348 ELECTROMAGNETIC OBSERVATIONS. [750.

The equation of motion is

(8)

— = C^secpe-^t^Pcosfa t + p). (9)

cl-t

,j a

When t = 0, 6 = 0, and — = C(dl = v.

dt

When <*>!$ + p = ->

Hence 0, = -- e v* cos/3. (11)

ME

JNow — — = or = o>i sec^/3, (12)

^4

x

tan£ = -, wj^^, (13)

7T jt-i

Hence ' *1 = , (l.)

which gives the first elongation in terms of the quantity of elec tricity in the transient current, and conversely, where T^ is the observed time of a single vibration as affected by the actual resist ance of damping. When A. is small we may use the approximate formula TT T

Method of Recoil.

750.] The method given above supposes the magnet to be at rest in its position of equilibrium when the transient current is passed through the coil. If we wish to repeat the experiment we must wait till the magnet is again at rest. In certain cases, however, in which we are able to produce transient currents of equal intensity, and to do so at any desired instant, the following method, described by Weber *, is the most convenient for making a continued series of observations.

• Rcsullate des Magnetisckcn Vereins, 1838, p. 98.

75O.] METHOD OF KECOIL. 349

Suppose that we set the magnet swinging by means of a transient current whose value is QQ. If, for brevity, we write

G V^TT2 -itan-i£

Jf~—T~~e n = jSr' (18)

then the first elongation

^ = KQ, = ^ (say). (19)

The velocity instantaneously communicated to the magnet at starting is jf Q

v-'^rft- (20)

When it returns through the point of equilibrium in a negative direction its velocity will be

v1 =—ve~^. (21)

The next negative elongation will be

6z = -61e-* = b1. (22)

When the magnet returns to the point of equilibrium, its velocity will be V2 = V()e-2\ (23)

Now let an instantaneous current, whose total quantity is — Q, be transmitted through the coil at the instant when the magnet is at the zero point. It will change the velocity v2 into v2— v, where

If Q is greater than Q0e~2^, the new velocity will be negative and equal to

^^ VH5 "BO*

The motion of the magnet will thus be reversed, and the next elongation will be negative,

03 = — K(Q — Q06~2A) = c1= —KQ + O^^. (25)

The magnet is then allowed to come to its positive elongation

and when it again reaches the point of equilibrium a positive current whose quantity is Q is transmitted. This throws the magnet back in the positive direction to the positive elongation

or, calling this the first elongation of a second series of four,

#2 = KQ (1 — <?"2A)-f a^e^K. (28)

Proceeding in this way, by observing two elongations + and — ,

then sending a positive current and observing two elongations

350

ELECTROMAGNETIC OBSERVATIONS.

[751-

— and -f , then sending a positive current, and so on, we obtain a series consisting of sets of four elongations, in each of which

and

(29)

(30)

If n series of elongations have been observed, then we find the logarithmic decrement from the equation

and Q from the equation

. (32)

Fig, 60.

The motion of the magnet in the method of recoil is graphically represented in Fig. 60, where the abscissa represents the time, and the ordinate the deflexion of the magnet at that time. See Art. 760.

Method of Multiplication.

751.] If we make the transient current pass every time that the

magnet passes through the zero point, and always so as to increase

the velocity of the magnet, then, if 01} 02, &c. are the successive

elongations, ^ = -KQ-e~* Olf (33)

Os=-KQ-e-^e2. (34)

The ultimate value to which the elongation tends after a great

many vibrations is found by putting 0n = — Qn-i > whence we find

(35)

If A is small, the value of the ultimate elongation may be large, but since this involves a long continued experiment, and a careful determination of A, and since a small error in A introduces a large error in the determination of Q, this method is rarely useful for

75I-] MISTIMING THE CURRENT. 351

numerical determination, and should be reserved for obtaining- evi dence of the existence or non-existence of currents too small to be observed directly.

In all experiments in which transient currents are made to act on the moving1 magnet of the galvanometer, it is essential that the whole current should pass while the distance of the magnet from the zero point remains a small fraction of the total elongation. The time of vibration should therefore be large compared with the time required to produce the current, and the operator should have his eye on the motion of the magnet, so as to regulate the instant of passage of the current by the instant of passage of the magnet through its point of equilibrium.

To estimate the error introduced by a failure of the operator to produce the current at the proper instant, we observe that the effect of a force in increasing the elongation varies as

and that this is a maximum when 0 = 0. Hence the error arising from a mistiming of the current will always lead to an under estimation of its value, and the amount of the error may be estimated by comparing the cosine of the phase of the vibration at the time of the passage of the current with unity.

CHAPTER XVII.

COMPARISON OF COILS.

Experimental Determination of the Electrical Constants of a Coil.

752.] WE have seen in Art. 717 that in a sensitive galvanometer the coils should he of small radius, and should contain many windings of the wire. It would he extremely difficult to determine the electrical constants of such a coil hy direct measurement of its form and dimensions, even if we could obtain access to every winding of the wire in order to measure it. But in fact the greater number of the windings are not only completely hidden by the outer windings, but we are uncertain whether the pressure of the outer windings may not have altered the form of the inner ones after the coiling of the wire.

It is better therefore to determine the electrical constants of the coil by direct electrical comparison with a standard coil whose con stants are known.

Since the dimensions of the standard coil must be determined by actual measurement, it must be made of considerable size, so that the unavoidable error of measurement of its diameter or circum ference may be as small as possible compared with the quantity measured. The channel in which the coil is wound should be of rectangular section, and the dimensions of the section should be small compared with the radius of the coil. This is necessary, not so much in order to diminish the correction for the size of the section, as to prevent any uncertainty about the position of those windings of the coil which are hidden by the external windings *.

• Large tangent galvanometers are sometimes made with a single circular con ducting ring of considerable thickness, which is sufficiently stiff to maintain its form without any support. This is not a good plan for a standard instrument. The dis tribution of the current within the conductor depends on the relative conductivity

753-] PRINCIPAL CONSTANTS OF A COIL. 353

The principal constants which we wish to determine are —

(1) The magnetic force at the centre of the coil due to a unit- current. This is the quantity denoted by G1 in Art. 700.

(2) The magnetic moment of the coil due to a unit-current. This is the quantity ff1 .

753.] To determine G1. Since the coils of the working galva nometer are much smaller than the standard coil, we place the galvanometer within the standard coil, so that their centres coincide, the planes of both coils being vertical and parallel to the earth's magnetic force. We have thus obtained a differential galvanometer one of whose coils is the standard coil, for which the value of G± is known, while that of the other coil is £/, the value of which we have to determine.

The magnet suspended in the centre of the galvanometer coil is acted on by the currents in both coils. If the strength of the current in the standard coil is y, and that in the galvanometer coil y', then, if these currents flowing in opposite directions produce a deflexion 6 of the magnet,

#tan8= G^y'-Gl7, (1)

where H is the horizontal magnetic force of the earth.

If the currents are so arranged as to produce no deflexion, we may find <?/ by the equation

<?/= -, e,. (2) We may determine the ratio of y to y in several ways. Since the value of Gl is in general greater for the galvanometer than for the standard coil, we may arrange the circuit so that the whole current y flows through the standard coil, and is then divided so that y' flows through the galvanometer and resistance coils, the combined resistance of which is J?13 while the remainder y — y' flows through another set of resistance coils whose combined resistance is E . of its various parts. Hence any concealed flaw in the continuity of the metal may cause the main stream of electricity to flow either close to the outside or close to the inside of the circular ring. Thus the true path of the current becomes uncertain. Besides this, when the current flows only once round the circle, especial care is necessary to avoid any action on the suspended magnet due to the current on its way to or from the circle, because the current in the electrodes is equal to that in the circle. In the construction of many instruments the action of this part of the current seems to have been altogether lost sight of. The most perfect method is to make one of the electrodes in the form of a metal tube, and the other a wire covered with insulating material, and placed inside the tube and concentric with it. The external action of the electrodes when thus arranged is zero, by Art. 683. VOL. II. A a 354 COMPARISON OF COILS. [754- We have then, by Art. 276, or = . (4) V H-i and G;=^+^Gl. (5) tf2 If there is any uncertainty about the actual resistance of the galvanometer coil (on account, say, of an uncertainty as to its tem perature) we may add resistance coils to it, so that the resistance of the galvanometer itself forms but a small part of Hlt and thus introduces but little uncertainty into the final result. 754.] To determine glt the magnetic moment of a small coil due to a unit-current flowing through it, the magnet is still suspended at the centre of the standard coil, but the small coil is moved parallel to itself along the common axis of both coils, till the same current, flowing in opposite directions round the coils, no longer deflects the magnet. If the distance between the centres of the coils is r, we have now £ =24 + 3^+4^f +&c. (6) ^.O ^.4 £>O By repeating the experiment with the small coil on the opposite side of the standard coil, and measuring the distance between the positions of the small coil, we eliminate the uncertain error in the determination of the position of the centres of the magnet and of the small coil, and we get rid of the terms in g2) g±, &c. If the standard coil is so arranged that we can send the current through half the number of windings, so as to give a different value to G19 we may determine a new value of r, and thus, as in Art. 454, we may eliminate the term involving g^ . It is often possible, however, to determine gz by direct measure ment of the small coil with sufficient accuracy to make it available in calculating the value of the correction to be applied to g^ in the equation i where #3 = — -ir0a(6«2-f 3f2 — 2»j2), by Art. 700. o Comparison of Coefficients of Induction. 755.] It is only in a small number of cases that the direct calculation of the coefficients of induction from the form and 755-] MUTUAL INDUCTION OF TWO COILS. 355 position of the circuits can be easily performed. In order to attain a sufficient degree of accuracy, it is necessary that the distance between the circuits should be capable of exact measurement. But when the distance between the circuits is sufficient to prevent errors of measurement from introducing large errors into the result, the coefficient of induction itself is necessarily very much reduced in magnitude. Now for many experiments it is necessary to make the coefficient of induction large, and we can only do so by bringing the circuits close together, so that the method of direct measure ment becomes impossible, and, in order to determine the coefficient of induction, we must compare it with that of a pair of coils ar ranged so that their coefficient may be obtained by direct measure ment and calculation. This may be done as follows : Let A and a be the standard pair of coils, B and b the coils to be compared with them. Con nect A and B in one circuit, and place the electrodes of the gal vanometer, G, at P and Q, so that the resistance of PAQ is R, and that of QBP is S, K being the resistance of the gal vanometer. Connect a and b in one circuit with the battery. Fig. 51. Let the current in A be », that in B, y> and that in the galvanometer, sc —y, that in the battery circuit being y. Then, if Ml is the coefficient of induction between A and «, and M2 that between B and b, the integral induction current through the galvanometer at breaking the battery circuit is x-y - y R" S 1 + (8) . R "" 8 By adjusting the resistances R and 8 till there is no current through the galvanometer at making or breaking the galvanometer circuit, the ratio of M2 to M1 may be determined by measuring that of S to R. A a 2 356 COMPARISON OF COILS. [756. Comparison of a Coefficient of Self-induction with a Coefficient of Mu tual Induction . 756.] In the branch AF of Wheatstone's Bridge let a coil be inserted, the coefficient of self-induc tion of which we wish to find. Let us call it L. In the connecting wire between A and the battery another coil is inserted. The coefficient of mutual induction be tween this coil and the coil in AF is M. It may be measured by the method described in Art. 755. If the current from A to F is #, and .p. 62 that from A to H is ^, that from Z to A, through B, will be oc+y. The external electromotive force from A to F is The external electromotive force along AH is A-H=Qy. (10) If the galvanometer placed between F and H indicates no current, either transient or permanent, then by (9) and (10), since I1 — F=0, whence L = - (l + ~) M. (13) ^o Since L is always positive, M must be negative, and therefore the current must flow in opposite directions through the coils placed in P and in B. In making the experiment we may either begin by adjusting the resistances so that PS=QR, (14) which is the condition that there may be no permanent current, and then adjust the distance between the coils till the galvanometer ceases to indicate a transient current on making and breaking the battery connexion ; or, if this distance is not capable of adjustment, we may get rid of the transient current by altering the resistances Q and S in such a way that the ratio of Q to S remains constant. If this double adjustment is found too troublesome, we may adopt 757-] SELF-INDUCTION. 357 a third method. Beginning with an arrangement in which the transient current due to self-induction is slightly in excess of that due to mutual induction, we may get rid of the inequality by in serting a conductor whose resistance is W between A and Z. The condition of no permanent current through the galvanometer is not affected by the introduction of W. We may therefore get rid of the transient current by adjusting the resistance of W alone. When this is done the value of L is . (15) . Comparison of the Coefficients of Self -induction of Two Coils. 757.] Insert the coils in two adjacent branches of Wheatstone's Bridge. Let L and N be the coefficients of self-induction of the coils inserted in P and in R respectively, then the condition of no galvanometer current is (P* + l^)8y=Qy(X* + N%), (16) whence PS = QJR, for no permanent current, (17) and — = — , for no transient current. (18) JT J-l/ Hence, by a proper adjustment of the resistances, both the per manent and the transient current can be got rid of, and then the ratio of L to N can be determined by a comparison of the resistances. CHAPTER XVIIL ELECTROMAGNETIC UNIT OF RESISTANCE. On the Determination of the Resistance of a Coil in Electro nic Measure. 758.] THE resistance of a conductor is defined as the ratio of the numerical value of the electromotive force to that of the current which it produces in the conductor. The determination of the value of the current in electromagnetic measure can be made by means of a standard galvanometer, when we know the value of the earth's magnetic force. The determination of the value of the electromotive force is more difficult, as the only case in which we can directly calculate its value is when it arises from the relative motion of the circuit with respect to a known magnetic system. 759.] The first determination of the resistance of a wire in electromagnetic measure was made by Kirchhoff*. He employed two coils of known form, A1 and A^ and calculated their coefficient of mutual induction from the geo metrical data of their form and position. These coils were placed in circuit with a galvanometer, 6r, and a battery, B, and two points of the circuit, P, between the coils, and Q, between the battery and galvanometer, were joined by the wire whose resistance, R, was to be measured. When the current is steady it is divided between the wire and the galvanometer circuit, and produces a certain permanent de flexion of the galvanometer. If the coil A1 is now removed quickly * * Bestimmong Her Constanten von welcher die Intensitat inducirter elektrischer Strome abhangt.' Pogg. Ann., Ixxvi (April 1849). 759-] KIRCHHOFF'S METHOD. 359 from A2 and placed in a position in which the coefficient of mutual induction between Al and A.2 is zero (Art. 538), a current of induc tion is produced in both circuits, and the galvanometer needle receives an impulse which produces a certain transient deflexion. The resistance of the wire, R, is deduced from a comparison between the permanent deflexion, due to the steady current, and the transient deflexion, due to the current of induction. Let the resistance of QGAl P be K, of PA2 £Q, B, and of PQ, R. Let Lj M and N be the coefficients of induction of Al and A2. Let x be the current in (7, and y that in J3, then the current from P to Q is x— y. Let E be the electromotive force of the battery, then )= o, (l) Rx + (B + R}y + -j- (Mx + Ny} = E. (2) When the currents are constant, and everything at rest, (K+R}x-Ry = 0. (3) If M now suddenly becomes zero on account of the separation of A1 from A2 , then, integrating with respect to t, J «/ "" \ / — Mx = lEdt = 0. (5) whence x = M(B\.R]^ ml ^2 ' (6) Substituting the value of y in terms of x from (3), we find 6 = ~R (B + R)(K+R}-R? (7) When, as in Kirchhoff 's experiment, both B and K are large compared with R, this equation is reduced to x _M ~x~~R' Of these quantities, x is found from the throw of the galvanometer due to the induction current. See Art. 768. The permanent cur rent, at, is found from the permanent deflexion due to the steady current; see Art. 746. M is found either by direct calculation from the geometrical data, or by a comparison with a pair of coils, for which this calculation has been made; see Art. 755. From 360 UNIT OF RESISTANCE. [760. these three quantities R can be determined in electromagnetic mea sure. These methods involve the determination of the period of vibra tion of the galvanometer magnet, and of the logarithmic decrement of its oscillations. Weber's Method by Transient Currents*. 760.] A coil of considerable size is mounted on an axle, so as to be capable of revolving about a vertical diameter. The wire of this coil is connected with that of a tangent galvanometer so as to form a single circuit. Let the resistance of this circuit be R. Let the large coil be placed with its positive face perpendicular to the magnetic meridian, and let it be quickly turned round half a revo lution. There will be an induced current due to the earth's mag netic force, and the total quantity of electricity in this current in electromagnetic measure will be where ffl is the magnetic moment of the coil for unit current, which in the case of a large coil may be determined directly, by mea suring the dimensions of the coil, and calculating the sum of the areas of its windings. If is the horizontal component of terrestrial magnetism, and R is the resistance of the circuit formed by the coil and galvanometer together. This current sets the magnet of the galvanometer in motion. If the magnet is originally at rest, and if the motion of the coil occupies but a small fraction of the time of a vibration of the magnet, then, if we neglect the resistance to the motion of the magnet, we have, by Art. 748, // T <2=^-2sinU (2) Cr 7T where G is the constant of the galvanometer, T is the time of vibration of the magnet, and 6 is the observed elongation. From these equations we obtain * = *°'¥15&' . (3) The value of H does not appear in this result, provided it is the same at the position of the coil and at that of the galvanometer. This should not be assumed to be the case, but should be tested by comparing the time of vibration of the same magnet, first at one of these places and then at the other. * ElcU. Moots*. ; or Pogg., Ann. Ixxxii, 337 (1851). 762.] WEBER'S METHOD. 361 761.] To make a series of observations Weber began with the coil parallel to the magnetic meridian. He then turned it with its positive face north, and observed the first elongation due to the negative current. He then observed the second elongation of the freely swinging magnet, and on the return of the magnet through the point of equilibrium he turned the coil with its positive face south. This caused the magnet to recoil to the positive side. The series Was continued as in Art. 750, and the result corrected for resistance. In this way the value of the resistance of the combined circuit of the coil and galvanometer was ascertained. In all such experiments it is necessary, in order to obtain suffi ciently large deflexions, to make the wire of copper, a metal which, though it is the best conductor, has the disadvantage of altering considerably in resistance with alterations of temperature. It is also very difficult to ascertain the temperature of every part of the apparatus. Hence, in order to obtain a result of permanent value from such an experiment, the resistance of the experimental circuit should be compared with that of a carefully constructed resistance- coil, both before and after each experiment. Weber's Method by observing the Decrement of the Oscillations of a Magnet. 762.] A magnet of considerable magnetic moment is suspended at the centre of a galvanometer coil. The period of vibration and the logarithmic decrement of the oscillations is observed, first with the circuit of the galvanometer open, and then with the circuit closed, and the conductivity of the galvanometer coil is deduced from the effect which the currents induced in it by the motion of the magnet have in resisting that motion. If T is the observed time of a single vibration, and A. the Na pierian logarithmic decrement for each single vibration, then, if we write ,, o> = ^> (1) and a = ~ , (2) the equation of motion of the magnet is of the form $ = Ce-atcos(o>t + (3}. (3) This expresses the nature of the motion as determined by observa tion. We must compare this with the dynamical equation of motion. 362 UNIT OF RESISTANCE. [?62. Let M be the coefficient of induction between the galvanometer coil and the suspended magnet. It is of the form M = Giffi Qi TO + $222 $2 W + &c., (4) where G1} G2, &c. are coefficients belonging to the coil, ffl3gz, &c. to the magnet, and Ql(0), Q.2(Q), &c., are zonal harmonics. of the angle between the axes of the coil and the magnet. See Art. 700. By a proper arrangement of the coils of the galvanometer, and by building up the suspended magnet of several magnets placed side by side at proper distances, we may cause all the terms of M after the first to become insensible compared with the first. If we also put (f> = -- 0, we may write M = Gm sin$, (5) where G is the principal coefficient of the galvanometer, m is the magnetic moment of the magnet, and $ is the angle between the axis of the magnet and the plane of the coil, which, in this ex periment, is always a small angle. If I/ is the coefficient of self-induction of the coil, and R its resistance, and y the current in the coil, 0, (6) or L~ -fj^y-f £mcos(£ - = 0. (7) U/t Cit The moment of the force with which the current y acts on the magnet is y —r— , or Gmy cos $. The angle </> is in this experiment ct cp so small, that we may suppose cos <£ = 1 . Let us suppose that the equation of motion of the magnet when the circuit is broken is where A is the moment of inertia of the suspended apparatus, S~- Cvv expresses the resistance arising from the viscosity of the air and of the suspension fibre, &c., and C<$> expresses the moment of the force arising from the earth's magnetism, the torsion of the sus pension apparatus, &c., tending to bring the magnet to its position of equilibrium. The equation of motion, as affected by the current, will be A + sc 762.] WEBER'S METHOD. 363 To determine the motion of the magnet, we have to combine this equation with (7) and eliminate y. The result is a linear differential equation of the third order. We have no occasion, however, to solve this equation, because the data of the problem are the observed elements of the motion of the magnet, and from these we have to determine the value of E. Let a0 and o)0 be the values of a and o> in equation (2) when the circuit is broken. In this case R is infinite, and the equation is reduced to the form (8). We thus find B=2AaQ, C=A(a^ + ^). (11) Solving equation (10) for R, and writing we find — o), where i=V — I, (12) Since the value of co is in general much greater than that of a, the best value of R is found by equating the terms in i o>, 0 2A(a — a0) a-a0 ' We may also obtain a value of R by equating the terms not involving i, but as these terms are small, the equation is useful only as a means of testing the accuracy of the observations. From these equations we find the following testing equation, (co2-o>02)2}. (15) Since LAv? is very small compared with G2m2, this equation a02-a2; (16) and equation (14) may be written E=GV_ L 2A(a-a0) r In this expression G may be determined either from the linear measurement of the galvanometer coil, or better, by comparison with a standard coil, according to the method of Art. 753. A is the moment of inertia of the magnet and its suspended apparatus, which is to be found by the proper dynamical method. o>, &>0, a and a0, are given by observation. 364 UNIT OF RESISTANCE. [763. The determination of the value of m, the magnetic moment of the suspended magnet, is the most difficult part of the investigation, because it is affected by temperature, by the earth's magnetic force, and by mechanical violence, so that great care must be taken to measure this quantity when the magnet is in the very same circum stances as when it is vibrating. The second term of R, that which involves L, is of less import ance, as it is generally small compared with the first term. The value of L may be determined either by calculation from the known form of the coil, or by an experiment on the extra-current of in duction. See Art. 756. Thomson's Method by a Revolving Coil. 763.] This method was suggested by Thomson to the Committee of the British Association on Electrical Standards, and the ex periment was made by M. M. Balfour Stewart, Fleeming Jenkin, and the author in 1863 *. A circular coil is made to revolve with uniform velocity about a vertical axis. A small magnet is suspended by a silk fibre at the centre of the coil. An electric current is induced in the coil by the earth's magnetism, and also by the suspended magnet. This current is periodic, flowing in opposite directions through the wire of the coil during different parts of each revolution, but the effect of the current on the suspended magnet is to produce a deflexion from the magnetic meridian in the direction of the rotation of the coil. 764.] Let H be the horizontal component of the earth's mag netism. Let y be the strength of the current in the coil.