327.] In a condenser of this kind, first charged in any way, next discharged through a wire of small resistance, and then insulated, no new electrification will appear. In most actual condensers, however, we find that after discharge and insulation a new charge is gradually developed, of the same kind as the original charge, but inferior in intensity. This is called the residual charge. To account for it we must admit that the constitution of the dielectric medium is different from that which we have just described. We shall find, however, that a medium formed of a conglomeration of small pieces of different simple media would possess this property.

Theory of a Composite Dielectric.

328.] We shall suppose, for the sake of simplicity, that the dielectric consists of a number of plane strata of different materials and of area unity, and that the electric forces act in the direction of the normal to the strata.

Let #15 az, &c. be the thicknesses of the different strata.

X1) X2, &c. the resultant electrical forces within the strata.

pl9 _p2, &c. the currents due to conduction through the strata. f>fi-> &c> the electric displacements.

ult «2, &c. the total currents, due partly to conduction and partly to variation of displacement.

STKATIFIEI) DIELECTRIC. 415

rlt r2 , &c. the specific resistances referred to unit of volume.

Klt K^ &c. the specific inductive capacities.

k^, kz, &c. the reciprocals of the specific inductive capacities.

E the electromotive force due to a voltaic battery, placed in the part of the circuit leading from the last stratum towards the first, which we shall suppose good conductors.

Q the total quantity of electricity which has passed through this part of the circuit up to the time t.

EQ the resistance of the battery with its connecting wires.

o-12 the surface-density of electricity on the surface which separates the first and second strata.

Then in the first stratum, we have, by Ohm's Law,

Il = rlp1. (1)

By the theory of electrical displacement,

XlL = ^klfl. (2)

By the definition of the total current,

' (3)

with similar equations for the other strata, in each of which the quantities have the suffix belonging to that stratum.

To determine the surface-density on any stratum, we have an equation of the form ^ = /2— /j, (4)

and to determine its variation we have

By differentiating (4) with respect to t, and equating the result to (5), we obtain

1-—* (•)

or, by taking account of (3),

«,_ = u2 =. &c. = u. (7)

That is, the total current u is the same in all the strata, and is equal to the current through the wire and battery. We have also, in virtue of equations (l) and (2), 1 1 dXl

(9)

from which we may find Xl by the inverse operation on u,

"

416 CONDUCTION IN DIELECTRICS. [329.

The total electromotive force E is

^=01J1 + «2I2 + &c., (10)

°r ~II

an equation between E, the external electromotive force, and u, the external current.

If the ratio of r to k is the same in all the strata, the equation reduces itself to

JT

». (12)

which is the case we have already examined, and in which, as we found, no phenomenon of residual charge can take place.

If there are n substances having different ratios of r to Jc, the general equation (11), when cleared of inverse operations, will be a linear differential equation, of the nth order with respect to E and of the (n— l)th order with respect to uf t being the independent variable.

From the form of the equation it is evident that the order of the different strata is indifferent, so that if there are several strata of the same substance we may suppose them united into one without altering the phenomena.

329.] Let us now suppose that at first f-^f^ &c. are all zero, and that an electromotive force E is suddenly made to act, and let us find its instantaneous effect.

Integrating (8) with respect to t, we find

q =Judt = ^-j'xidt+ _-L-.rl + const. (13)

Now, since X: is always in this case finite, / X-^dt must be in-

sensible when t is insensible, and therefore, since Xx is originally zero, the instantaneous effect will be

. (14)

Hence, by equation (10),

^=477 (Vl + V2 + &O & (15)

and if C be the electric capacity of the system as measured in this instantaneous way,

329.] ELECTRIC 'ABSORPTION/ 417

This is the same result that we should have obtained if we had neglected the conductivity of the strata.

Let us next suppose that the electromotive force E is continued uniform for an indefinitely long time, or till a uniform current of conduction equal to p is established through the system.

We have then X1 = r^p^ etc., and therefore by (10),

= fa « i + r2 a2 + &c.)j3. ( 1 7)

If It be the total resistance of the system,

E

R = — = rx «x + r2 «2 + &c. (18)

In this state we have by (2),

so that ^=(>- (19)

If we now suddenly connect the extreme strata by means of a conductor of small resistance, E will be suddenly changed from its original value 2$0 to zero, and a quantity Q of electricity will pass through the conductor.

To determine Q we observe that if X± be the new value of X1, then by (13), X/ = Xl + 4 TT^ Q. (20)

Hence, by (10), putting E = 0,

0 = 01X1 + &c. + 47r(a1£1 + 02£2 + &c.)Q, (21)

or 0 = E0+Q. (22)

Hence Q = — CEQ where C is the capacity, as given by equation (IB). The instantaneous discharge is therefore equal to the in- stantaneous charge.

Let us next suppose the connexion broken immediately after this discharge. We shall then have u — 0, so that by equation (8),

47T*! ,

Zi = re ^ , (23)

where X' is the initial value after the discharge. Hence, at any time t,

The value of E at any time is therefore

VOL. I. EC

418 CONDUCTION IN DIELECTEICS. [330.

and the instantaneous discharge after any time t is EG. This is called the residual discharge.

If the ratio of r to li is the same for all the strata, the value of E will be reduced to zero. If, however, this ratio is not the same, let the terms be arranged according to the values of this ratio in descending order of magnitude.

The sum of all the coefficients is evidently zero, so that when t = 0, E = 0. The coefficients are also in descending order of magnitude, and so are the exponential terms when t is positive. Hence, when t is positive, E will be positive, so that the residual discharge is always of the same sign as the primary discharge.

When t is indefinitely great all the terms disappear unless any of the strata are perfect insulators, in which case r-^ is infinite for that stratum, and R is infinite for the whole system, and the final value of E is not zero but

E = E^l-iTia^C). (25)

Hence, when some, but not all, of the strata are perfect insulators, a residual discharge may be permanently preserved in the system.

330.] We shall next determine the total discharge through a wire of resistance RQ kept permanently in connexion with the extreme strata of the system, supposing the system first charged by means of a long-continued application of the electromotive force E.

At any instant we have

E = alr1pl + a2r2p2 + 8cc.+RQu = 0, (26)

and also, by (3), U=p1+-~j±. (27)

Hence (R + A>) « = «i *i f^ + «2 '2 ^ + &c. (28)

Integrating with respect to t in order to find Q, we get

(R + £„)$ = a, r, (/I'-/,) + a, r, (//-/2) + &c., (29)

where/! is the initial, and// the final value of/i-

In this case//= 0, and by (2) and (20) /t =

Hence (S + ^Q = + + &c. -E.CR, (30)

where the summation is extended to all quantities of this form belonging to every pair of strata.

331-] RESIDUAL DISCHARGE. 419

It appears from this that Q is always negative, that is to say, in the opposite direction to that of the current employed in charging the system.

This investigation shews that a dielectric composed of strata of different kinds may exhibit the phenomena known as electric absorption and residual discharge, although none of the substances of which it is made exhibit these phenomena when alone. An investigation of the cases in which the materials are arranged otherwise than in strata would lead to similar results, though the calculations would be more complicated, so that we may conclude that the phenomena of electric absorption may be ex- pected in the case of substances composed of parts of different kinds, even though these individual parts should be microscopically small.

It by no means follows that every substance which exhibits this phenomenon is so composed, for it may indicate a new kind of electric polarization of which a homogeneous substance may be capable, and this in some cases may perhaps resemble electro- chemical polarization much more than dielectric polarization.

The object of the investigation is merely to point out the true mathematical character of the so-called electric absorption, and to shew how fundamentally it differs from the phenomena of heat which seem at first sight analogous.

331.] If we take a thick plate of any substance and heat it on one side, so as to produce a flow of heat through it, and if we then suddenly cool the heated side to the same temperature as the other, and leave the plate to itself, the heated side of the plate will again become hotter than the other by conduction from within.

Now an electrical phenomenon exactly analogous to this can be produced, and actually occurs in telegraph cables, but its mathe- matical laws, though exactly agreeing with those of heat, differ entirely from those of the stratified condenser.

In the case of heat there is true absorption of the heat into the substance with the result of making it hot. To produce a truly analogous phenomenon in electricity is impossible, but we may imitate it in the following way in the form of a lecture- room experiment.

Let Al , A2 , &c. be the inner conducting surfaces of a series of condensers, of which i?0, -Z?15 £2t &c. are the outer surfaces.

Let Alt A2, &c. be connected in series by connexions of rcsist-

420

CONDUCTION IN DIELECTRICS.

ance Et and let a current be passed along this series from left to right.

Let us first suppose the plates _Z?0, JBlt J52, each insulated and free from charge. Then the total quantity of electricity on each of the plates JB must remain zero, and since the electricity on the plates A is in each case equal and opposite to that of the opposed

Fig. 26.

surface they will not be electrified, and no alteration of the current will be observed.

But let the plates IB be all connected together, or let each be connected with the earth. Then, since the potential of Al is positive, while that of the plates B is zero, A1 will be positively electrified and Z?x negatively.

If P19 P2, &c. are the potentials of the plates Al} A2, &c.} and C the capacity of each, and if we suppose that a quantity of electricity equal to QQ passes through the wire on the left, Qi through the connexion Rlt and so on, then the quantity which exists on the plate A-L is Q0— Q13 and we have

«.-«!= 43-

Similarly &-&= GA>

and so on.

But by Ohm's Law we have

If we suppose the values of C the same for each plate, and those of R the same for each wire, we shall have a series of equations of the form

332.] THEORY OF ELECTKIC CABLES. 421

If there are n quantities of electricity to be determined, and if either the total electromotive force, or some other equivalent con- ditions be given, the differential equation for determining1 any one of them will be linear and of the nih order.

By an apparatus arranged in this way, Mr. Varley succeeded in imitating the electrical action of a cable 12,000 miles long.

When an electromotive force is made to act along the wire on the left hand, the electricity which flows into the system is at first principally occupied in charging the different condensers beginning with Alt and only a very small fraction of the current appears at the right hand till a considerable time has elapsed. If galvano- meters be placed in circuit at R-^ R2, &c. they will be affected by the current one after another, the interval between the times of equal indications being greater as we proceed to-the right.

332.] In the case of a telegraph cable the conducting wire is separated from conductors outside by a cylindrical sheath of gutta- percha, or other insulating material* Each portion, of the cable thus becomes a condenser, the outer surface of which is always at potential zero. Hence, in a given portion of the cable, the quantity of free electricity at the surface of the conducting- wire is equal to the product of the potential into the capacity of the portion of the cable considered as a condenser.

If a13 a2 are the outer and inner radii of the insulating sheath, and if K is its specific dielectric capacity, the capacity of unit of length of the cable is, by Art. 126,

Let v be the potential at any point of the wire, which we may consider as the same at every part of the same section.

Let Q be the total quantity of electricity which has passed through that section since the beginning of the current. Then the quantity which at the time t exists between sections at x and at is . dO v d

or -

and this is, by what we have said, equal to cvbx.

422 CONDUCTION IN DIELECTRICS. [333.

Hence " = ~ 5T (2)

Again, the electromotive force at any section is — -j-, and by Ohm's Law, dv . dQ

~s=*ir (3)

where k is the resistance of unit of length of the conductor, and —- is the strength of the current. Eliminating Q between (2) and (3), we find c^=^. (4)

This is the partial differential equation which must be solved in order to obtain the potential at any instant at any point of the cable. It is identical with that which Fourier gives to determine the temperature at any point of a stratum through which heat is flowing in a direction normal to the stratum. In the case of heat c represents the capacity of unit of volume, or -what Fourier denotes by CD, and k represents the reciprocal of the conductivity.

If the sheath is not a perfect insulator, and if k± is the resist- ance of unit of length of the sheath to conduction through it in a radial direction, then if pL is the specific resistance of the insulating material, it is easy to shew that

The equation (2) will no longer be true, for the electricity is expended not only in charging the wire to the extent represented

by cv, but in escaping at a rate represented by — . Hence the rate of expenditure of electricity will be

I

whence, by comparison with (3), we get

dv d2v k

. .

and this is the equation of conduction of heat in a rod or ring as given by Fourier *.

333.] If we had supposed that a body when raised to a high potential becomes electrified throughout its substance as if elec- tricity were compressed into it, we should have arrived at equa- tions of this very form. It is remarkable that Ohm himself,

• Theorie de la Ckaleur, Art. 105.

334-]

HYDROSTATICAL ILLUSTRATION.

423

-A -

-o -

misled by the analogy between electricity and heat, entertained an opinion of this kind, and was thus, by means of an erroneous opinion, led to employ the equations of Fourier to express the true laws of conduction of electricity through a long wire, long before the real reason of the appropriateness of these equations had been suspected.

Mechanical Illustration of the Properties of a Dielectric.

334.] Five tubes of equal sectional area A, B, C, D and P are

arranged in circuit as in the figure.

A, B, C and D are vertical and equal, jr f*0 P *5^\

and P is horizontal.

The lower halves of A, B, C, D are filled with mercury, their upper halves and the horizontal tube P are filled with water.

A tube with a stopcock Q con- nects the lower part of A and B with that of C and _Z>, and a piston P is made to slide in the horizontal tube.

Let us begin by supposing that the level of the mercury in the four tubes is the same, and that it is indicated by A^ BQ, <70, D0, that the piston is at P0, and that the stopcock Q is shut.

Now let the piston be moved from P0 to Plt a distance a. Then, since the sections of all the tubes are equal, the level of the mercury in A and C will rise a distance a, or to Al and Clt and the mercury in B and D will sink an equal distance #, or to Bl and D± .

The difference of pressure on the two sides of the piston will be represented by 4 a.

This arrangement may serve to represent the state of a dielectric acted on by an electromotive force 4 a.

The excess of water in the tube D may be taken to represent a positive charge of electricity on one side of the dielectric, and the excess of mercury in the tube A may represent the negative charge on the other side. The excess of pressure in the tube P on the side of the piston next D will then represent the excess of potential on the positive side of the dielectric.

(

^1 • 1^

~\

• C -

1

-v

Bo

•v

-v

-*,:

Q

Fig. 27.

424 CONDUCTION IN DIELECTRICS, [334-

If the piston is free to move it will move back to P0 and be in equilibrium there. This represents the complete discharge of the dielectric.

During the discharge there is a reversed motion of the liquids throughout the whole tube, and this represents that change of electric displacement which we have supposed to take place in a dielectric.

I have supposed every part of the system of tubes filled with incompressible liquids, in order to represent the property of all electric displacement that there is no real accumulation of elec- tricity at any place.

Let us now consider the effect of opening the stopcock Q while the piston P is at Pt.

The level of A± and DL will remain unchanged, but that of B and C will become the same, and will coincide with JB0 and C0 .

The opening of the stopcock Q corresponds to the existence of a part of the dielectric which has a slight conducting power, but which does not extend through the whole dielectric so as to form an open channel.

The charges on the opposite sides of the dielectric remain in- sulated, but their difference of potential diminishes.

In fact, the difference of pressure on the two sides of the pi&ton sinks from \a to 2 a during the passage of the fluid through Q.

If we now shut the stopcock Q and allow the piston P to move freely, it will come to equilibrium at a point P2 , and the discharge will be apparently only half of the charge.

The level of the mercury in A and B will be \a above its original level, and the level in the tubes C and D will be \a below its original level. This is indicated by the levels A2, _Z?2,

C2, A-

If the piston is now fixed and the stopcock opened, mercury will flow from £ to C till the level in the two tubes, is again at JBQ and C0. There will then be a difference of pressure == a on the two sides of the piston P. If the stopcock is then closed and the piston P left free to move, it will again come to equilibrium at a point PB , half way between P2 and P0 . This corresponds to the residual charge which is observed when a charged dielectric is first dis- charged and then left to itself. It gradually recovers part of its charge, and if this is again discharged a third charge is formed, the successive charges diminishing in quantity. In the case of the illustrative experiment each charge is half of the preceding, and the

334-] HYDROSTATICAL ILLUSTRATION. 425

discharges, which are J, J, &c. of the original charge, form a series whose sum is equal to the original charge.

If, instead of opening and closing the stopcock, we had allowed it to remain nearly, but not quite, closed during the whole experiment, we should have had a case resembling that of the electrification of a dielectric which is a perfect insulator and yet exhibits the pheno- menon called ' electric absorption.'

To represent the case in which there is true conduction through the dielectric we must either make the piston leaky, or we must establish a communication, between the top of the tube A and the top of the tube D.

In this way we may construct a mechanical illustration of the properties of a dielectric of any kind, in which the two electricities are represented by two real fluids, and the electric potential is represented by fluid pressure. Charge and discharge are repre- sented by the motion of the piston P, and electromotive force by the resultant force on the piston.

CHAPTEK XT.

THE MEASUREMENT OF ELECTRIC RESISTANCE.

335.] IN the present state of electrical science, the determination of the electric resistance of a conductor may be considered as the cardinal operation in electricity, in the same sense that the deter- mination of weight is the cardinal operation in chemistry.

The reason of this is that the determination in absolute measure of other electrical magnitudes, such as quantities of electricity, electromotive forces, currents, &c., requires in each case a com- plicated series of operations, involving generally observations of time, measurements of distances, and determinations of moments of inertia, and these operations, or at least some of them, must be repeated for every new determination, because it is impossible to preserve a unit of electricity, or of electromotive force, or of current, in an unchangeable state, so as to be available for direct comparison.

But when the electric resistance of a properly shaped conductor of a properly chosen material has been once determined, it is found that it always remains the same for the same temperature, so that the conductor may be used as a standard of resistance, with which that of other conductors can be compared, and the comparison of two resistances is an operation which admits of extreme accuracy.

When the unit of electrical resistance has been fixed on, material copies of this unit, in the form of ' Resistance Coils,5 are prepared for the use of electricians, so that in every part of the world electrical resistances may be expressed in terms of the same unit. These unit resistance coils are at present the only examples of material electric standards which can be preserved, copied, and used for the purpose of measurement. Measures of electrical capacity, which are also of great importance, are still defective, on account of the disturbing influence of electric absorption.

336.] The unit -of resistance may be an entirely arbitrary one, as in the case of Jacobi's Etalon, which was a certain copper wire of 22.4932 grammes weight, 7.61975 metres length, and 0.667

339-] STANDARDS OF RESISTANCE. 427

millimetres diameter. Copies of this have been made by Leyser of Leipsig, and are to be found in different places.

According to another method the unit may be defined as the resistance of a portion of a definite substance of definite dimensions. Thus, Siemens' unit is defined as the Tesistance of a column of mercury of one metre long-, and one square millimetre section, at the temperature 0°C.

337.] Finally, the unit may be defined with reference to the electrostatic or the electromagnetic system of units. In practice the electromagnetic system is used in all telegraphic operations, and therefore the only systematic units actually in use are those of this system.

In the electromagnetic system, as we shall shew at the proper place, a resistance is a quantity homogeneous with a velocity, and may therefore be expressed as a velocity. See Art. 628.

338.] The first actual measurements on this system were made by Weber, who employed as his unit one millimetre per second. Sir W. Thomson afterwards used one foot per second as a unit, but a large number of electricians have now agreed to use the unit of the British Association, which professes to represent a resistance which, expressed *as a velocity, is ten millions of metres per second. The magnitude of this unit is -more convenient than that of Weber's unit, which is too small. It is sometimes referred to as the B.A. unit, but in order to connect it with the name of the discoverer of the iaws of resistance, it is called the Ohm.

339.] To recollect its value in absolute measure it is useful to know that ten millions of metres is professedly the distance from the pole to the equator, measured along the meridian of Paris. A body, therefore, which in one second travels along a meridian from the pole to the equator would have a velocity which, on the electromagnetic system, is professedly represented by an Ohm.

I say professedly, because, if more accurate researches should prove that the Ohm, as constructed from the British Association's material standards, is not really represented by this velocity, elec- tricians would not alter their standards, but would apply a cor- rection. In the same way the metre is professedly one ten-millionth of a certain quadrantal arc, but though this is found not to be exactly true, the length of the metre has not been altered, but the dimensions of the earth are expressed by a less simple number.

According to the system of the British Association, the absolute value of the unit is originally chosen so as to represent as nearly

428

MEASUREMENT OF RESISTANCE.

[340.

as possible a quantity derived from the electromagnetic absolute system.

340.] When a material unit representing this abstract quantity has been made, other standards are constructed by copying this unit, a process capable of extreme accuracy — of much greater accuracy than, for instance, the copying of foot-rules from a standard foot.

These copies, made of the most permanent materials, are dis- tributed over all parts of the world, so that it is not likely that any difficulty will be found in obtaining copies of them if the original standards should be lost.

But such units as that of Siemens can without very great labour be reconstructed with considerable accuracy, so that as the relation of the Ohm to Siemens unit is known, the Ohm can be reproduced even without having a standard to copy, though the labour is much greater and the accuracy much less than by the method of copying.

Finally, the Ohm may be reproduced by the electromagnetic method by which it was originally determined. This method, which is considerably more laborious than the determination of a foot from the seconds pendulum, is probably inferior in accuracy to that last mentioned. On the other hand, the determination of the electromagnetic unit in terms of the Ohm with an amount of accuracy corresponding to the progress of electrical science, is a most important physical research and well worthy of being repeated.

The actual resistance coils constructed to represent the Ohm were made of an alloy of two parts of silver and one of pla- tinum in the form of wires. from .5 milli- metres to .8 millimetres diameter, and from one to two metres in length. These wires were soldered to stout copper electrodes. The wire itself was covered with two layers of silk, imbedded in solid paraffin, and. enclosed in a thin brass case, so that it can be easily brought to a temperature at which its resistance is accurately one Ohm. This temperature is marked on the insulating support of the coil. (See Fig. 28.)

Fig. 28.

341-] RESISTANCE COILS. 429

On the Forms of Resistance Coils.

341 .] A Resistance Coil is a conductor capable of being easily placed in the voltaic circuit, so as to introduce into the circuit a known resistance.

The electrodes or ends of the coil must be such that no appre- ciable error may arise from the mode of making the connexions. For resistances of considerable magnitude it is sufficient that the electrodes should be made of stout copper wire or rod well amal- gamated with mercury at the ends, and that the ends should be made to press on flat amalgamated copper surfaces placed in mercury cups.

For very great resistances it is sufficient that the electrodes should be thick pieces of brass, and that the connexions should be made by inserting a wedge of brass or copper into the interval between them. This method is found very convenient.

The resistance coil itself consists of a wire well covered with silk, the ends of which are soldered permanently to the elec- trodes.

The coil must be so arranged that its temperature may be easily observed. For this purpose the wire is coiled on a tube and covered with another tube, so that it may be placed in a vessel of water, and that the water may have access to the inside and the outside of the coil.

To avoid the electromagnetic effects of the current in the coil the wire is first doubled back on itself and then coiled on the tube, so that at every part of the coil there are equal and opposite currents in the adjacent parts of the wire.

When it is desired to keep two coils at the same temperature the wires are sometimes placed side by side and coiled up together. This method is especially useful when it is more important to secure equality of resistance than to know the absolute value of the resistance, as in the case of the equal arms of Wheatstone's Bridge, (Art. 347).

When measurements of resistance were first attempted, a resist- ance coil, consisting of an uncovered wire coiled in a spiral groove round a cylinder of insulating material, was much used. It was called a Rheostat. The accuracy with which it was found possible to compare resistances was soon found to be inconsistent with the use of any instrument in which the contacts are not more perfect than can be obtained in the rheostat. The rheostat, however, is

430

MEASUREMENT OF RESISTANCE,

[342.

still used for adjusting the resistance where accurate measurement is not required.

Resistance coils are generally made of those metals whose resist- ance is greatest and which vary least with temperature. German silver fulfils these conditions very well, but some specimens are found to change their properties during the lapse of years. Hence, for standard coils, several pure metals, and also an alloy of platinum and silver, have been employed, and the relative resistance of these during several years has been, found constant up to the limits of modern accuracy.

342.] For very great resistances, such as several millions of Ohms, the wire must be either very long or very thin, and the construction of the coil is expensive and difficult. Hence tellurium and selenium have been proposed as materials for constructing standards of great resistance. A very ingenious and easy method of construction has been lately proposed by Phillips *. On a piece of ebonite or ground glass a fine pencil-line is drawn. The ends of this filament of plumbago are connected to metallic electrodes, and the whole is then covered with insulating varnish. If it should be found that the resistance of such a pencil-line remains constant, this will be the best method of obtaining a resistance of several millions of Ohms.

343.] There are various arrangements by which resistance coils may be easily introduced into a circuit.

For instance, a series of coils of which the resistances are 1, 2, 4, 8, 16, &c., arranged according to the powers of 2, may be placed in a box in series.

The electrodes consist of stout brass plates, so arranged on the outside of the box that by inserting a brass plug or wedge between

• Phil Mag., July, 1870.

344-]

RESISTANCE BOXES.

431

two of them as a shunt, the resistance of the corresponding coil may be put out of the circuit. This arrangement was introduced by Siemens.

Each interval between the electrodes is marked with the resist- ance of the corresponding coil, so that if we wish to make the resistance box equal to 107 we express 107 in the binary scale as 64 + 32 + 8 + 2 + 1 or 1101011. We then take the plugs out of the holes corresponding to 64, 32, 8, 2 and 1, and leave the plugs in 16 and 4.

This method, founded on the binary scale, is that in which the smallest number of separate coils is needed, and it is also that which can be most readily tested. For if we have another coil equal to 1 we can test the equality of 1 and l', then that of 1 + 1' and 2, then that of 1 + 1' + 2 and 4, and so on.

The only disadvantage of the arrangement is that it requires a familiarity with the binary scale of notation, which is not generally possessed by those accustomed to express every number in the decimal scale.

344.] A box of resistance coils may be arranged in a different way for the purpose of mea- suring conductivities instead of resistances.

The coils are placed so that one end of each is connected with a long thick piece of metal which forms one elec- trode of the box, and the other

Fig. 30.

end is connected with a stout piece of brass plate as in the former case.

The other electrode of the box is a long brass plate, such that by inserting brass plugs between it and the electrodes of the coils it may be connected to the first electrode through any given set of coils. . The conductivity of the box is then the sum of the con- ductivities of the coils.

In the figure, in which the resistances of the coils are 1, 2, 4, &c., and the plugs are inserted at 2 and 8, the conductivity of the box is 1 + 1- = f , and the resistance of the box is therefore £ or 1.6.

This method of combining resistance coils for the measurement of fractional resistances was introduced by Sir W. Thomson under the name of the method of multiple arcs. See Art. 276.

324

MEASUREMENT OF EESISTANCE.

[345-

On the Comparison of Resistances.

345.] If E is the electromotive force of a battery, and R the resistance of the battery and its connexions, including the galvan- ometer used in measuring the current, and if the strength of the current is 7 when the battery connexions are closed, and I13 72 when additional resistances rlt r2 are introduced into the circuit, then, by Ohm's

Eliminating U, the electromotive force of the battery, and R the resistance of the battery and its connexions, we get Ohm's formula ^ _ (7-7T)72

This method requires a measurement of the ratios of /, 7t and 72, and this implies a galvanometer graduated for absolute mea- surements.