A Treatise on Electricity and Magnetism, Vol. 2 (1873) — part 11 of 27

The quantities a', (3', y are sometimes called the determinants of the circuit / referred to the point P. Their resultant is called by Ampere the directrix of the electrodynamic action.

It is evident from the equation, that the force whose components

dX dY . dZ .

are -^—> -=-, and ~ is perpendicular both to ds and to this as as ds

directrix, and is represented numerically by the area of the parallel ogram whose sides are ds and the directrix.

5 1 9-] FORCE BETWEEN TWO FINITE CURRENTS. 157

In the language of quaternions, the resultant force on ds is the vector part of the product of the directrix multiplied by ds.

Since we already know that the directrix is the same thing as the magnetic force due to a unit current in the circuit /, we shall henceforth speak of the directrix as the magnetic force due to the circuit.

518.] We shall now complete the calculation of the components of the force acting between two finite currents, whether closed or open.

Let p be a new function of r, such that

"oo

P = i/ (B-C)dr, (24)

then by (17) and (20)

d% d

and equations (11) become

& — - ' ' au&(Q+P)

B ** , ** \

O = — ^7-7 > O = — - •

ds ds J

With these values of the component forces, equation (13) becomes

l_L_ I' _UL . (27}

w ds ds' ds' ds

519.] Let

F = I Ipds, G = I mpds, H = I npds, (28)

i/O JQ JQ

F = f'l'p ds', G' = f'm'pds', H'= [''n'pds'. (29)

^0 "'0 Jo

These quantities have definite values for any given point of space. When the circuits are closed, they correspond to the components of the vector-potentials of the circuits.

Let L be a new function of r, such that

fr

L — I r(Q + p)dr, (30)

^o

and let M be the double integral

M = I I pcosedsds', (31)

«^0 * 0

158 AMPERE'S THEORY. [520.

which, when the circuits are closed, becomes their mutual potential, then (27) may be written

«•„•* \dM dL }

dsds'~ dsds I da dx^ \

520.] Integrating1, with respect to s and *', between the given limits, we find

d_ dx dx

• F'P-FfA-FP, + FAf, (33)

where the subscripts of L indicate the distance, r, of which the quantity L is a function, and the subscripts of F and F' indicate the points at which their values are to be taken.

The expressions for Y and Z may be written down from this. Multiplying the three components by dxt dy, and dz respectively, we obtain

Xdx+Ydy + Zdz = DM-D(Lpp,—LAP,—LA,p

X — — — =- (LPp> — LAP'— LA' p

P,-Ay, (34)

where D is the symbol of a complete differential.

Since Fdx + Gdy + Hdz is not in general a complete differential of a function of #,y, £, Xdx + Ydy + Zdz is not a complete differential for currents either of which is not closed.

521.] If, however, both currents are closed, the terms in I/, F, G, H, F, G't H' disappear, and

Xdx+Ydy + Zdz = DM, (35)

where M is the mutual potential of two closed circuits carrying unit currents. The quantity M expresses the work done by the electro magnetic forces on either conducting circuit when it is moved parallel to itself from an infinite distance to its actual position. Any alteration of its position, by which M is increased, will be assisted by the electromagnetic forces.

It may be shewn, as in Arts. 490, 596, that when the motion of the circuit is not parallel to itself the forces acting on it are still determined by the variation of M, the potential of the one circuit on the other.

522.] The only experimental fact which we have made use of in this investigation is the fact established by Ampere that the action of a closed current on any portion of another current is perpendicular to the direction of the latter. Every other part of

524.] HIS FORMULA. 159

the investigation depends on purely mathematical considerations depending on the properties of lines in space. The reasoning there fore may be presented in a much more condensed and appropriate form by the use of the ideas and language of the mathematical method specially adapted to the expression of such geometrical relations — the Quaternions of Hamilton.

This has been done by Professor Tait in the Quarterly Mathe matical Journal, 1866, and in his treatise on Quaternions, § 399, for Ampere's original investigation, and the student can easily adapt the same method to the somewhat more general investigation given here.

523.] Hitherto we have made no assumption with respect to the quantities A, B, C, except that they are functions of r, the distance between the elements. We have next to ascertain the form of these functions, and for this purpose we make use of Ampere's fourth case of equilibrium, Art. 508, in which it is shewn that if all the linear dimensions and distances of a system of two circuits be altered in the same proportion, the currents remaining the same, the force between the two circuits will remain the same.

Now the force between the circuits for unit currents is -= — , and

dos

since this is independent of the dimensions of the system, it must be a numerical quantity. Hence M itself, the coefficient of the mutual potential of the circuits, must be a quantity of the dimen sions of a line. It follows, from equation (31), that p must be the reciprocal of a line, and therefore by (24), B— (7 must be the inverse square of a line. But since B and C are both functions of r, B—C must be the inverse square of r or some numerical multiple of it.

524.] The multiple we adopt depends on our system of measure ment. If we adopt the electromagnetic system, so called because it agrees with the system already established for magnetic measure ments, the value of M ought to coincide with that of the potential of two magnetic shells of strength unity whose boundaries are the two circuits respectively. The value of M in that case is, by

Art. 423, /"/"cos* ,

M = J I - ds ds', (36)

the integration being performed round both circuits in the positive direction. Adopting this as the numerical value of M, and com paring with (31), we find

p = , and S-C=~. (37)

160 AMPERE'S THEORY. [525-

525.] We may now express the components of the force on ds arising from the action of ds' in the most general form consistent with experimental facts.

The force on ds is compounded of an attraction

1 /dr dr d2r \ . d2 0 . 7 , 1

R = -H- l-y- -=-, — 2r -7-77) ^^ dsds' -f r -^—j-,11 ds ds rz ^ds ds dsds' dsds

in the direction of r,

S = 77 i i'ds ds in the direction of ds,

as

and S' = —^ ii'ds ds' in the direction of ds'. ds /NO

where Q — / Cdr, and since C is an unknown function of r, we

J r

know only that Q is some function of r.

526.] The quantity Q cannot be determined, without assump tions of some kind, from experiments in which the active current forms a closed circuit. If we suppose with Ampere that the action between the elements ds and ds' is in the line joining them, then S and 8' must disappear, and Q must be constant, or zero. The force is then reduced to an attraction whose value is

(39)

Ampere, who made this investigation long before the magnetic system of units had been established, uses a formula having a numerical value half of this, namely

1 A dr dr dr N . ., _

R = -2 (- -7- -T7 - r -j—r^Jjds ds'. (40)

f2 \9 fix Of d*njfJ** v '

Here the strength of the current is measured in what is called electro dynamic measure. If i, i' are the strength of the currents in electromagnetic measure, and j, j' the same in electrodynamic mea sure, then it is plain that

jf = 2ii', or j = ^i. (41)

Hence the unit current adopted in electromagnetic measure is greater than that adopted in electrodynamic measure in the ratio of «/2 to 1.

The only title of the electrodynamic unit to consideration is that it was originally adopted by Ampere, the discoverer of the law of action between currents. The continual recurrence of <s/2 in calculations founded on it is inconvenient, and the electro magnetic system has the great advantage of coinciding numerically

527.] FOUK ASSUMPTIONS. 161

with all our magnetic formulae. As it is difficult for the student to bear in mind whether he is to multiply or to divide by /2, we shall henceforth use only the electromagnetic system, as adopted by Weber and most other writers.

Since the form and value of Q have no effect on any of the experiments hitherto made, in which the active current at least is always a closed one, we may, if we please, adopt any value of Q which appears to us to simplify the formulae.

Thus Ampere assumes that the force between two elements is in the line joining them. This gives Q = 0,

(42) r

Grassmann * assumes that two elements in the same straight line have no mutual action. This gives

Q 1 R- 3 d*T 8- l dr 8'- l - (43)

V= 2r' ~Trdsds" 2r* els'3 ~ 2r* ds ( }

We might, if we pleased, assume that the attraction between two elements at a given distance is proportional to the cosine of the angle between them. In this case

„ 1 _ 1 0 1 dr 0, 1 dr , . .

«=--> JZ = ^«»c, * = -F5p. S'=^Ts. (44)

Finally, we might assume that the attraction and the oblique forces depend only on the angles which the elements make with the line joining them, and then we should have

0- 2 R- 3ldrdr S- 2- S'-*~. (45)

V* ~P * VS3?1 ~PdS> ~ r* ds ( }

527.] Of these four different assumptions that of Ampere is undoubtedly the best, since it is the only one which makes the forces on the two elements not only equal and opposite but in the straight line which joins them.

• Pogg., Ann. Ixiv. p. 1 (1845).

VOL. II. M

CHAPTER III

ON THE INDUCTION OF ELECTRIC CURRENTS.

528.] THE discovery by Orsted of the magnetic action of an electric current led by a direct process of reasoning to that of magnetization by electric currents, and of the mechanical action between electric currents. It was not, however, till 1831 that Faraday, who bad been for some time endeavouring to produce electric currents by magnetic or electric action, discovered the con ditions of magneto-electric induction. The method which Faraday employed in his researches consisted in a constant appeal to ex periment as a means of testing the truth of his ideas, and a constant cultivation of ideas under the direct influence of experiment. In his published researches we find these ideas expressed in language which is all the better fitted for a nascent science, because it is somewhat alien from the style of physicists who have been accus tomed to established mathematical forms of thought.

The experimental investigation by which Ampere established the laws of the mechanical action between electric currents is one of the most brilliant achievements in science.

The whole, theory and experiment, seems as if it had leaped, full grown and full armed, from the brain of the ' Newton of elec tricity.' It is perfect in form, and unassailable in accuracy, and it is summed up in a formula from which all the phenomena may be deduced, and which must always remain the cardinal formula of electro-dynamics.

The method of Ampere, however, though cast into an inductive form, does not allow us to trace the formation of the ideas which guided it. We can scarcely believe that Ampere really discovered the law of action by means of the experiments which he describes. We are led to suspect, what, indeed, he tells us himself*, that he

• Theorie des Phenomenes Elect rodynamiqucs, p. 9.

529.] ' FARADAY'S SCIENTIFIC METHOD. 163

discovered the law by some process which he has not shewn us, and that when he had afterwards built up a perfect demon stration he removed all traces of the scaffolding by which he had raised it.

Faraday, on the other hand, shews us his unsuccessful as well as his successful experiments, and his crude ideas as well as his developed ones, and the reader, however inferior to him in inductive power, feels sympathy even more than admiration, and is tempted to believe that, if he had the opportunity, he too would be a dis coverer. Every student therefore should read Ampere's research as a splendid example of scientific style in the statement of a dis covery, but he should also study Faraday for the cultivation of a scientific spirit, by means of the action and reaction which will take place between newly discovered facts and nascent ideas in his own mind.

It was perhaps for the advantage of science that Faraday, though thoroughly conscious of the fundamental forms of space, time, and force, was not a professed mathematician. He was not tempted to enter into the many interesting researches in pure mathematics which his discoveries would have suggested if they had been exhibited in a mathematical form, and he did not feel called upon either to force his results into a shape acceptable to the mathe matical taste of the time, or to express them in a form which mathematicians might attack. He was thus left at leisure to do his proper work, to coordinate his ideas with his facts, and to express them in natural, untechnical language.

It is mainly with the hope of making these ideas the basis of a mathematical method that I have undertaken this treatise.

529.] We are accustomed to consider the universe as made up of parts, and mathematicians usually begin by considering a single par ticle, and then conceiving its relation to another particle, and so on. This has generally been supposed the most natural method. To conceive of a particle, however, requires a process of abstraction, since all our perceptions are related to extended bodies, so that the idea of the all that is in our consciousness at a given instant is perhaps as primitive an idea as that of any individual thing. Hence there may be a mathematical method in which we proceed from the whole to the parts instead of from the parts to the whole. For example, Euclid, in his first book, conceives a line as traced out by a point, a surface as swept out by a line, and a solid as generated by a surface. But he also defines a surface as the

M 2

164 MAGNETO-ELECTRIC INDUCTION,* [530.

boundary of a solid, a line as the edge of a surface, and a point as the extremity of a line.

In like manner we may conceive the potential of a material system as a function found by a certain process of integration with respect to the masses of the bodies in the field, or we may suppose these masses themselves to have no other mathematical meaning

than the volume-integrals of — V2^? where ^ is the potential.

In electrical investigations we may use formulae in which the quantities involved are the distances of certain bodies, and the electrifications or currents in these bodies, or we may use formulae which involve other quantities, each of which is continuous through all space.

The mathematical process employed in the first method is in tegration along lines, over surfaces, and throughout finite spaces, those employed in the second method are partial differential equa tions and integrations throughout all space.

The method of Faraday seems to be intimately related to the second of these modes of treatment. He never considers bodies as existing with nothing between them but their distance, and acting on one another according to some function of that distance. He conceives all space as a field of force, the lines of force being in general curved, and those due to any body extending from it on all sides, their directions being modified by the presence of other bodies. He even speaks * of the lines of force belonging to a body as in some sense part of itself, so that in its action on distant bodies it cannot be said to act where it is not. This, however, is not a dominant idea with Faraday. I think he would rather have said that the field of space is full of lines of force, whose arrangement depends on that of the bodies in the field, and that the mechanical and electrical action on each body is determined by the lines which abut on it.

PHENOMENA OF MAGNETO-ELECTRIC INDUCTION f.

530.] 1. Induction by Variation of the Primary Current.

Let there be two conducting circuits, the Primary and the Secondary circuit. The primary circuit is connected with a voltaic

• Exp. Res., ii. p. 293 ; iii. p. 447.

t Read Faraday's Experimental Researches, series i and ii.

530.] ELEMENTARY PHENOMENA. 165

battery by which the primary current may be produced, maintained, stopped, or reversed. The secondary circuit includes a galvano meter to indicate any currents which may be formed in it. This galvanometer is placed at such a distance from all parts of the primary circuit that the primary current has no sensible direct influence on its indications.

Let part of the primary circuit consist of a straight wire, and part of the secondary circuit of a straight wire near, and parallel to the first, the other parts of the circuits being at a greater distance from each other.

It is found that at the instant of sending a current through the straight wire of the primary circuit the galvanometer of the secondary circuit indicates a current in the secondary straight wire in the opposite direction. This is called the induced current. If the primary current is maintained constant, the induced current soon disappears, and the primary current appears to produce no effect on the secondary circuit. If now the primary current is stopped, a secondary current is observed, which is in the same direction as the primary current. Every variation of the primary current produces electromotive force in the secondary circuit. When the primary current increases, the electromotive force is in the opposite direction to the current. When it diminishes, the electromotive force is in the same direction as the current. When the primary current is constant, there is no electromotive force.

These effects of induction are increased by bringing the two wires nearer together. They are also increased by forming them into two circular or spiral coils placed close together, and still more by placing an iron rod or a bundle of iron wires inside the coils.

2. Induction ~by Motion of the Primary Circuit.

We have seen that when the primary current is maintained constant and at rest the secondary current rapidly disappears.

Now let the primary current be maintained constant, but let the primary straight wire be made to approach the secondary straight wire. During the approach there will be a secondary current in the opposite direction from the primary.

If the primary circuit be moved away from the secondary, there will be a secondary current in the same direction as the primary.

3. Induction by Motion of the Secondary Circuit. If the secondary circuit be moved, the secondary current is

166 MAGNETO-ELECTRIC INDUCTION. [S31-

opposite to the primary when the secondary wire is approaching- the primary wire, and in the same direction when it is receding- from it.

In all cases the direction of the secondary current is such that the mechanical action between the two conductors is opposite to the direction of motion, being a repulsion when the wires are ap proaching, and an attraction when they are receding. This very important fact was established by Lenz *.

4. Induction by the Relative Motion of a Magnet and the Secondary

Circuit.

If we substitute for the primary circuit a magnetic shell, whose edge coincides with the circuit, whose strength is numerically equal to that of the current in the circuit, and whose austral face cor responds to the positive face of the circuit, then the phenomena produced by the relative motion of this shell and the secondary circuit are the same as those observed in the case of the primary circuit.

531.] The whole of these phenomena may be summed up in one law. When the number of lines of magnetic induction which pass through the secondary circuit in the positive direction is altered, an electromotive force acts round the circuit, which is measured by the rate of decrease of the magnetic induction through the circuit.

532.] For instance, let the rails of a railway be insulated from the earth, but connected at one terminus through a galvanometer, and let the circuit be completed by the wheels and axle of a rail way carriage at a distance x from the terminus. Neglecting the height of the axle above the level of the rails, the induction through the secondary circuit is due to the vertical component of the earth's magnetic force, which in northern latitudes is directed downwards. Hence, if b is the gauge of the railway, the horizontal area of the circuit is bx, and the surface-integral of the magnetic induction through it is Zbxt where Z is the vertical component of the magnetic force of the earth. Since Z is downwards, the lower face of the circuit is to be reckoned positive, and the positive direction of the circuit itself is north, east, south, west, that is, in the direction of the sun's apparent diurnal course.

Now let the carriage be set in motion, then x will vary, and

• Pogg., Ann. xxi. 483 (1834.)

533-] DIRECTION OF THE FORCE. 167

there will be an electromotive force in the circuit whose value

„, das is — Zb -=-.

dt

If x is increasing, that is, if the carriage is moving away from the terminus, this electromotive force is in the negative direction, or north, west, south, east. Hence the direction of this force through the axle is from right to left. If x were diminishing, the absolute direction of the force would be reversed, but since the direction of the motion of the carriage is also reversed, the electro motive force on the axle is still from right to left, the observer in the carriage being always supposed to move face forwards. In southern latitudes, where the south end of the needle dips, the electromotive force on a moving body is from left to right.

Hence we have the following rule for determining the electro motive force on a wire moving through a field of magnetic force. Place, in imagination, your head and feet in the position occupied by the ends of a compass needle which point north and south respec tively ; turn your face in the forward direction of motion, then the electromotive force due to the motion will be from left to right.

533.] As these directional relations are important, let us take another illustration. Suppose a metal girdle laid round the earth at the equator, and a metal wire laid along the meridian of Green wich from the equator to the north pole. /

Let a great quadrantal arch of r/A metal be constructed, of which one extremity is pivoted on the north pole, while the other is carried round the equator, sliding on the great girdle of the earth, and following the sun in his daily course. There will then be an electromotive force along the moving quadrant, acting from the pole towards the equator.

The electromotive force will be the same whether we suppose the earth at rest and the quadrant moved from east to west, or whether we suppose the quadrant at rest and the earth turned from west to east. If we suppose the earth to rotate, the electromotive force will be the same whatever be the form of the part of the circuit fixed in space of which one end touches one of the pole&

168 MAGNETO-ELECTRIC INDUCTION. [534-

and the other the equator. The current in this part of the circuit is from the pole to the equator.

The other part of the circuit, which is fixed with respect to the earth, may also be of any form, and either within or without the earth. In this part the current is from the equator to either pole.

534.] The intensity of the electromotive force of magneto -electric induction is entirely independent of the nature of the substance of the conductor in which it acts, and also of the nature of the conductor which carries the inducing current.

To shew this, Faraday * made a conductor of two wires of different metals insulated from one another by a silk covering, but twisted together, and soldered together at one end. The other ends of the wires were connected with a galvanometer. In this way the wires were similarly situated with respect to the primary circuit, but if the electromotive force were stronger in the one wire than in the other it would produce a current which would be indicated by the galvanometer. He found, however, that such a combination may be exposed to the most powerful electromotive forces due to in duction without the galvanometer being affected. He also found that whether the two branches of the compound conductor consisted of two metals, or of a metal and an electrolyte, the galvanometer was not affected f.

Hence the electromotive force on any conductor depends only on the form and the motion of that conductor, together with the strength, form, and motion of the electric currents in the field.

535.] Another negative property of electromotive force is that it has of itself no tendency to cause the mechanical motion of any body, but only to cause a current of electricity within it.

If it actually produces a current in the body, there will be mechanical action due to that current, but if we prevent the current from being formed, there will be no mechanical action on the body itself. If the body is electrified, however, the electro motive force will move the body, as we have described in Electro statics.

536.] The experimental investigation of the laws of the induction of electric currents in fixed circuits may be conducted with considerable accuracy by methods in which the electromotive force, and therefore the current, in the galvanometer circuit is rendered zero.

For instance, if we wish to shew that the induction of the coil

• Rrp. fas., 195. f Ib., 200.

536.]

EXPERIMENTS OF COMPARISON.

169

A on the coil X is equal to that of B upon Y, we place the first pair of coils A and X at a sufficient distance from the second pair

Fig. 32.

£ and Y. We then connect A and B with a voltaic battery, so that we can make the same primary current flow through A in the positive direction and then through B in the negative direction. We also connect X and Y with a galvanometer, so that the secondary current, if it exists, shall flow in the same direction through X and Yin series.

Then, if the induction of A on X is equal to that of B on Y, the galvanometer will indicate no induction current when the battery circuit is closed or broken.

The accuracy of this method increases with the strength of the primary current and the sensitiveness of the galvanometer to in stantaneous currents, and the experiments are much more easily performed than those relating to electromagnetic attractions, where the conductor itself has to he delicately suspended.

A very instructive series of well devised experiments of this kind is described by Professor Felici of Pisa *.

I shall only indicate briefly some of the laws which may be proved in this way.

(1) The electromotive force of the induction of one circuit on another is independent of the area of the section of the conductors and of the material of which they are made.

For we can exchange any one of the circuits in the experiment for another of a different section and material, but of the same form, without altering the result.

• Annettes dc Chimie, xxxiv. p. G6 (1852), and Nuovo Cimento, ix. p. 345 (1859).

170 MAGNETO-ELECTRIC INDUCTION. [537-

(2) The induction of the circuit A on the circuit X is equal to that of X upon A.

For if we put A in the galvanometer circuit, and X in the battery circuit, the equilibrium of electromotive force is not disturbed.

(3) The induction is proportional to the inducing current.

For if we have ascertained that the induction of A on X is equal to that of B on Y, and also to that of C on Z, we may make the battery current first flow through A, and then divide itself in any proportion between B and C. Then if we connect X reversed, Y and Z direct, all in series, with the galvanometer, the electromotive force in X will balance the sum of the electromotive forces in Y

(4) In pairs of circuits forming systems geometrically similar the induction is proportional to their linear dimensions.

For if the three pairs of circuits above mentioned are all similar, but if the linear dimension of the first pair is the sum of the corresponding linear dimensions of the second and third pairs, then, if A, B, and C are connected in series with the battery, and X reversed, Y and Z also in series with the galvanometer, there will be equilibrium.

(5) The electromotive force produced in a coil of n windings by a current in a coil of m windings is proportional to the product mn.

537.] For experiments of the kind we have been considering the galvanometer should be as sensitive as possible, and its needle as light as possible, so as to give a sensible indication of a very small transient current. The experiments on induction due to motion require the needle to have a somewhat longer period of vibration, so that there may be time to effect certain motions of the conductors while the needle is not far from its position of equilibrium. In the former experiments, the electromotive forces in the galvanometer circuit were in equilibrium during the whole time, so that no current passed through the galvano meter coil. In those now to be described, the electromotive forces act first in one direction and then in the other, so as to produce in succession two currents in opposite directions through the gal vanometer, and we have to shew that the impulses on the galvano meter needle due to these successive currents are in certain cases equal and opposite.

The theory of the application of the galvanometer to the measurement of transient currents will be considered more at length in Art. 748. At present it is sufficient for our purpose to ob-

538-J FELICl's EXPERIMENTS. 171

serve that as long- as the galvanometer needle is near its position of equilibrium the deflecting force of the current is proportional to the current itself, and if the whole time of action of the current is small compared with the period of vibration of the needle, the final velocity of the magnet will be proportional to the total quantity of electricity in the current. Hence, if two currents pass in rapid succession, conveying equal quantities of electricity in opposite directions, the needle will be left without any final velocity.

Thus, to shew that the induction-currents in the secondary circuit, due to the closing and the breaking of the primary circuit, are equal in total quantity but opposite in direction, we may arrange the primary circuit in connexion with the battery, so that by touching a key the current may be sent through the primary circuit, or by removing the finger the contact may be broken at pleasure. If the key is pressed down for some time, the galvanometer in the secondary circuit indicates, at the time of making contact, a transient current in the opposite direction to the primary current. If contact be maintained, the induction current simply passes and disappears. If we now break contact, another transient current passes in the opposite direction through the secondary circuit, and the galvanometer needle receives an impulse in the opposite direction.

But if we make contact only for an instant, and then break contact, the two induced currents pass through the galvanometer in such rapid succession that the needle, when acted on by the first current, has not time to move a sensible distance from its position of equilibrium before it is stopped by the second, and, on account of the exact equality between the quantities of these transient currents, the needle is stopped dead.

If the needle is watched carefully, it appears to be jerked suddenly from one position of rest to another position of rest very near the first.

In this way we prove that the quantity of electricity in the induction current, when contact is broken, is exactly equal and opposite to that in the induction current when contact is made.

538.] Another application of this method is the following, which is given by Felici in the second series of his Researches.

It is always possible to find many different positions of the secondary coil I>, such that the making or the breaking of contact in the primary coil A produces no induction current in 7?. The

172 MAGNETO-ELECTKIC INDUCTION. [539-

positions of the two coils are in such cases said to be conjugate to each other.

Let BI and B2 be two of these positions. If the coil B be sud denly moved from the position B± to the position J32, the algebraical sum of the transient currents in the coil B is exactly zero, so that the galvanometer needle is left at rest when the motion of B is completed.

This is true in whatever way the coil B is moved from Bl to B2^ and also whether the current in the primary coil A be continued constant, or made to vary during the motion.

Again, let B' be any other position of B not conjugate to A, so that the making or breaking of contact in A produces an in duction current when B is in the position B'.

Let the contact be made when B is in the conjugate position _Z?1? there will be no induction current. Move B to B'> there will be an induction current due to the motion, but if B is moved rapidly to B', and the primary contact then broken, the induction current due to breaking contact will exactly annul the effect of that due to the motion, so that the galvanometer needle will be left at rest. Hence the current due to the motion from a conjugate position to any other position is equal and opposite to the current due to breaking contact in the latter position.

Since the effect of making contact is equal and opposite to that of breaking it, it follows that the effect of making contact when the coil B is in any position B' is equal to that of bringing the coil from any conjugate position Bl to B' while the current is flowing through A.