A Treatise on Electricity and Magnetism, Vol. 1 (1881) — part 27 of 28

If the resistances r^ and r2 are equal, then 21 and 72 are equal, and we can test the equality of currents by a galvanometer which is not capable of determining their ratios.

But this is rather to be taken as an example of a faulty method than as a practical method of determining resistance. The electro- motive force E cannot be maintained rigorously constant, and the internal resistance of the battery is also exceedingly variable, so that any methods in which these are assumed to be even for a short time constant are not to be depended on.

346.] The comparison of resistances can be made with extreme

accuracy by either of two methods, in which the result is in- dependent of variations of R and JE.

346.] COMPARISON OP RESISTANCES. 433

The first of these methods depends on the use of the differential galvanometer, an instrument in which there are two coils, the currents in which are independent of each other, so that when the currents are made to flow in opposite directions they act in opposite directions on the needle, and when the ratio of these currents is that of m to n they have no resultant effect on the galvanometer needle.

Let II , 72 be the currents through the two coils of the galvan- ometer, then the deflexion of the needle may be written

8 = m!1—nI2.

Now let the battery current I be divided between the coils of the galvanometer, and let resistances A and B be introduced into the first and second coils respectively. Let the remainder of the resistance of the coils and their connexions be a and ft respect- ively, and let the resistance of the battery and its connexions between C and D be r, and its electromotive force E.

Then we find, by Ohm's Law, for the difference of potentials between C and D,

and since

7? + /3 ET

-- _- _

where D =

The deflexion of the galvanometer needle is therefore

and if there is no observable deflexion, then we know that the quantity enclosed in brackets cannot differ from zero by more than a certain small quantity, depending on the power of the battery, the suitableness of the arrangement, the delicacy of the galvano- meter, and the accuracy of the observer.

Suppose that B has been adjusted so that there is no apparent deflexion.

Now let another conductor A' be substituted for A^ and let A' be adjusted till there is no apparent deflexion. Then evidently to a first approximation A'= A.

To ascertain the degree of accuracy of this estimate, let the altered quantities in the second observation be accented, then

VOL. i. r f

434 MEASUKEMENT OF EESISTANCE. [346.

~

Hence n(A'-A) = b-~b'.

If b and S', instead of being both apparently zero, had been only observed to be equal, then, unless we also could assert that E = W^ the right-hand side of the equation might not be zero. In fact, the method would be a mere modification of that already described.

The merit of the method consists in the fact that the thing observed is the absence of any deflexion, or in other words, the method is a Null method, one in which the non-existence of a force is asserted from an observation in which the force, if it had been different from zero by more than a certain small amount, would have produced an observable effect.

Null methods are of great value where they can be employed, but they can only be employed where we can cause two equal and opposite quantities of the same kind to enter into the experiment together.

In the case before us both b and b' are quantities too small to be observed, and therefore any change in the value of E will not affect the accuracy of the result.

The actual degree of accuracy of this method might be ascer- tained by making a number of observations in each of which A' is separately adjusted, and comparing the result of each observation with the mean of the whole series.

But by putting A' out of adjustment by a known quantity, as, for instance, by inserting at A or at B an additional resistance equal to a hundredth part of A or of JB, and then observing the resulting deviation of the galvanometer needle, we can estimate the number of degrees corresponding to an error of one per cent. To find the actual degree of precision we must estimate the smallest deflexion which could not escape observation, and compare it with the deflexion due to an error of one per cent.

• If the comparison is to be made between A and -8, and if the positions of A and JB are exchanged, then the second equation becomes

• This investigation is taken from Weber's treatise on Galvanometry. Gottingen Transactions, x. p. 65.

DIFFERENTIAL GALVANOMETER. 435

) = ~b'y whence (m + n) (B—A) = ~ b - ^ 6'.

£j £j

If ^ and n, A and .#, a and /3 are approximately equal, then

Here 5-5' may be taken to be the smallest observable deflexion of the galvanometer.

If the galvanometer wire be made longer and thinner, retaining the same total mass, then n will vary as the length of the wire and a as the square of the length. Hence there will be a minimum value of M + ')^ + « + 2r) when

If we suppose r, the battery resistance, small compared with A, this gives a=$A;

or, the resistance of each coil of the galvanometer should oe one-third of the resistance to lie measured. We then find 2

If we allow the current to flow through one only of the coils of the galvanometer, and if the deflexion thereby produced is A (supposing the deflexion strictly proportional to the deflecting force), then

mE 3nE .„ 1 .

A =s -T- -- = - — if / = 0 and a = - A. A + a + r 4 A 3

Hence

In the differential galvanometer two currents are made to produce equal and opposite effects on the suspended needle. The force with which either current acts on the needle depends not only on the strength of the current, but on the position of the windings of the wire with respect to the needle. Hence, unless the coil is very carefully wound, the ratio of m to n may change when the position of the needle is changed, and therefore it is necessary to determine this ratio by proper methods during each

i fa

436 MEASUREMENT OF RESISTANCE, [347.

course of experiments if any alteration of the position of the needle is suspected.

The other null method, in which Wheatstone's Bridge is used, requires only an ordinary galvanometer, and the observed zero deflexion of the needle is due, not to the opposing action of two currents, but to the non-existence of a current in the wire. Hence we have not merely a null deflexion, but a null current as the phenomenon observed, and no errors can arise from want of regularity or change of any kind in the coils of the galvanometer. The galvanometer is only required to be sensitive enough to detect the existence and direction of a current, without in any way determining its value or comparing its value with that of another current.

347.] Wheatstone's Bridge consists essentially of six conductors connecting four points. An electromotive force E is made to act between two of the points by means of a voltaic battery in- troduced between B and C. The current between the other two points 0 and A is measured by a galvanometer.

Under certain circumstances this current becomes zero. The conductors BC and OA are then said to be conjugate to each other, which implies a certain relation between the resistances of the other four conductors, and this relation is made use of in measuring resistances.

If the current in OA is zero, the potential at 0 must be equal to that at A. Now when we know the potentials at B and C we can determine those at 0 and A by the rule given in Art. 275, provided there is no current in OA,

.

^x _

whence the condition is

where I, c, /3, y are the resistances in CA, AS, BO, and OC re- spectively.

To determine the degree of accuracy attainable by this method we must ascertain the strength of the current in OA when this condition is not fulfilled exactly.

Let A, B, C and 0 be the four points. Let the currents along BC, CA and AB be so, y and z, and the resistances of these

WHEATSTONE'S BRIDGE.

437

conductors 0, b and c. Let the currents along OA, OB and OC be £ r;, £ and the resistances a, (3 and y. Let an electromotive force E act along BC. Required the current £ along OA.

Let the potentials at the points At B, C and 0 be denoted by the symbols A, B, C and 0. The equations of conduction are

cz = A— B, with the equations of continuity

= 0— C\

-* = o,

— x = 0,

— y = 0.

By considering the system as made up of three circuits OBC, OCA and OAB, in which the currents are #, y, 2 respectively, and applying Kirchhoff's rule to each cycle, we eliminate the values of the potentials 0, A, J3, C, and the currents f, r/, £ and obtain the following equations for x, y and zt

—yx

Hence, if we put

— az

=0,

-ay

— y

—a

-a

we find

and

»= -

348.] The value of D may be expressed in the symmetrical form,

or, since we suppose the battery in the conductor a and the galvanometer in a, we may put B the battery resistance for a and G the galvanometer resistance for a. We then find

If the electromotive force E were made to act along OA, the resistance of OA being still a, and if the galvanometer were placed

438 MEASUREMENT OF RESISTANCE. [349-

in EC, the resistance of BC being still #, then the value of D would remain the same, and the current in BC due to the electro- motive force E acting along OA would be equal to the current in OA due to the electromotive force E acting in BC.

But if we simply disconnect the battery and the galvanometer, and without altering their respective resistances connect the battery to 0 and A and the galvanometer to B and C, then in the value of D we must exchange the values of B and G. If I/ be the value of D after this exchange, we find D-V = G-

Let us suppose that the resistance of the galvanometer is greater than that of the battery.

Let us also suppose that in its original position the galvanometer connects the junction of the two conductors of least resistance /3, y with the junction of the two conductors of greatest resistance b, c, or, in other words, we shall suppose that if the quantities b, c, y, /3 are arranged in order of magnitude, t> and c stand together, and y and ft stand together. Hence the quantities &— (3 and c—y are of the same sign, so that their product is positive, and therefore D— fi' is of the same sign as B— G.

If therefore the galvanometer is made to connect the junction of the two greatest resistances with that of the two least, and if the galvanometer resistance is greater than that of the battery, then the value of D will be less, and the value of the deflexion of the galvanometer greater, than if the connexions are exchanged.

The rule therefore for obtaining the greatest galvanometer de- flexion in a given system is as follows :

Of the two resistances, that of the battery and that of the galvanometer, connect the greater resistance so as to join the two greatest to the two least of the four other resistances.

349.] We shall suppose that we have to determine the ratio of the resistances of the conductors AB and AC, and that this is to be done by finding a point 0 on the conductor BOC, such that when the points A and 0 are connected by a wire, in ihe course of which a galvanometer is inserted, no sensible deflexion of the galvano- meter needle occurs when the battery is made to act between B and C.

The conductor BOG may be supposed to be a wire of uniform resistance divided into equal parts, so that the 'ratio of the resist- ances of BO and OC may be read off at once.

349-] WHEATSTONE'S BRIDGE. 439

Instead of the whole conductor being a uniform wire, we may make the part near 0 of such a wire, and the parts on each side may be coils of any form, the resistance of which is accurately known.

We shall now use a different notation instead of the symmetrical notation with which we commenced.

Let the whole resistance of BAC be R.

Let c = mR and b — (l—m) R.

Let the whole resistance of BOG be S.

Let /3 = nS and y = (1 — n) S.

The value of n is read off directly, and that of m is deduced from it when there is no sensible deviation of the galvanometer.

Let the resistance of the battery and its connexions be B, and that of the galvanometer and its connexions G.

We find as before

—2mn) BRS, and if f is the current in the galvanometer wire t ERS .

(=—(n-m).

In order to obtain the most accurate results we must make the deviation of the needle as great as possible compared with the value of (n — m). This may be done by properly choosing the dimensions of the galvanometer and the standard resistance wire.

It will be shewn, when we come to Galvanometry, Art. 716, that when the form of a galvanometer wire is changed while its mass remains constant, the deviation of the needle for unit current is proportional to the length, but the resistance increases as the square of the length. Hence the maximum deflexion is shewn to occur when the resistance of the galvanometer wire is equal to the constant resistance of the rest of the circuit.

In the present case, if 6 is the deviation,

where C is some constant, and G is the galvanometer resistance which varies as the square of the length of the wire. Hence we find that in the value of Dt when 8 is a maximum, the part involving G must be made equal to the rest of the expression.

If we also put m = n, as is the case if we have made a correct observation, we find the best value of G to be

440

MEASUREMENT OF RESISTANCE.

[350.

This result is easily obtained by considering the resistance from A to 0 through* the system, remembering that £C, being conjugate to AO, has no effect on this resistance.

In the same way we. should find that if the total area of the acting surfaces of the battery is given, the most advantageous arrangement of the battery is when

Finally, we shall determine the value of 8 such that a given change in the value of n may produce the greatest galvanometer deflexion. By differentiating the expression for f we find

BE "

If we have a great many determinations of resistance to make in which the actual resistance has nearly the same value, then it may be worth while to prepare a galvanometer and a battery for this purpose. In this case we find that the best arrangement is

and if n = i G= \R.

On the Use of WheaUtonJs Bridge.

350.] We have already explained the general theory of Wheat- stone's Bridge, we shall now consider some of its applications.

Fig. 33.

The comparison which can be effected with the greatest exact- ness is that of two equal resistances.

35o.] USE OF WHEATSTONE'S BRIDGE. 441

Let us suppose that {3 is a standard resistance coil, and that we wish to adjust y to be equal in resistance to /3. '

Two other coils, b and c, are prepared which are equal or nearly equal to each other, and the four coils are placed with their electrodes in mercury cups so that the current of the battery is divided between two branches, one consisting of (3 and y and the other of b and c. The coils b and c are connected by a wire PR, as uniform in its resistance as possible, and furnished with a scale of equal parts.

The galvanometer wire connects the junction of (3 and y with a point Q of the wire PR, and the point of contact at Q is made to vary till on closing first the battery circuit and then the galvanometer circuit, no deflexion of the galvanometer needle is observed.

The coils /3 and y are then made to change places, and a new position is found for Q. If this new position is the same as the old one, then we know that the exchange of /3 and y has produced no change in the proportions of the resistances, and therefore y is rightly adjusted. If Q has to be moved, the direction and amount of the change will indicate the nature and amount of the alteration of the length of the wire of y, which will make its resistance equal to that of /3.

If the resistances of the coils b and c, each including part of the wire PR up to its zero reading, are equal to that of b and c divisions of the wire respectively, then, if x is the scale reading of Q in the first case, and y that in the second, e + a? _/3 c+y __ y

b—x ~ y ' b—y ~~ /3*

2 whence = 1

Since £— y is nearly equal to c + x, and both are great with respect to x or y, we may write this

and

"When y is adjusted as well as we can, we substitute for b and c other coils of (say) ten times greater resistance.

The remaining difference between /3 and y will now produce a ten times greater difference in the position of Q than with the

442

MEASUREMENT OF RESISTANCE.

[351.

original coils I and c, and in this way we can continually increase the accuracy of the comparison.

The adjustment by means of the wire with sliding contact piece is more quickly made than by means of a resistance box, and it is capable of continuous variation.

The battery must never be introduced instead of the galvano- meter into the wire with a sliding contact, for the passage of a powerful current at the point of contact would injure the surface of the wire. Hence this arrangement is adapted for the case in which the resistance of the galvanometer is greater than that of the battery.

When y, the resistance to be measured, a the resistance of the battery, and a the resistance of the galvanometer, are given, the best values of the other resistances have been shewn by Mr. Oliver Heaviside (Phil. Mag. Feb. 1873) to be

c = */aa,

a-f y

On the Measurement of Small 'Resistances.

351.} When a short and thick conductor is introduced into a circuit its resistance is so small compared with the resistance occasioned by unavoidable faults in the connexions, such as want of contact or imperfect soldering, that no correct value of the

resistance can be deduced from experi- ments made in the way described above.

The object of such experiments is generally to determine the specific re- sistance of the substance, and it is re- sorted to in cases when the substance cannot be obtained in the form of a long thin wire, or when the resistance to transverse as well as to longitudinal conduction has to be measured. Sir W. Thomson* has described a method applicable to such cases, which we may take as an example of a system of nine conductors.

35i.] THOMSON'S METHOD FOR SMALL RESISTANCES. 443

The most important part of the method consists in measuring the resistance, not of the whole length of the conductor, but of the part between two marks on the conductor at some little dis- tance from its ends.

The resistance which we wish to measure is that experienced by a current whose intensity is uniform in any section of the conductor, and which flows in a direction parallel to its axis. Now close to the extremities, when the current is introduced by means of electrodes, either soldered, amalgamated, or simply pressed to the ends of the conductor, there is generally a want of uniformity in the distribution of the current in the conductor. At a short distance from the extremities the current becomes

Fig. 35.

sensibly uniform. The student may examine for himself the investigation and the diagrams of Art. 193, where a current is introduced into a strip of metal with parallel sides through one of the sides, but soon becomes itself parallel to the sides.

The resistances of the conductors between certain marks S, S' and T, T' are to be compared.

The conductors are placed in series, and with connexions as perfectly conducting as possible, in a battery circuit of small resist- ance. A wire SVT is made to touch the conductors at S and T, and S' VT' is another wire touching them at S' and T'.

The galvanometer wire connects the points ^and V of these wires.

The wires SFT and S'V'T are of resistance so great that the resistance due to imperfect connexion at S, T, S' or T' may be neglected in comparison with the resistance of the wire, and 7t V are taken so that the resistances in the branches of either wire leading to the two conductors are nearly in the ratio of the resist- ances of the two conductors.

Calling^ and F the resistances of the conductors SS' and T'T. A and C those of the branches 87 sail VT.

444

MEASUREMENT OF RESISTANCE.

[352.

Calling P and R those of the branches 8' V and V'T'. „ Q that of the connecting piece S'T'. „ £ that of the battery and its connexions. „ G that of the galvanometer and its connexions. The symmetry of the system may be understood from the skeleton diagram. Fig. 34.

The condition that B the battery and G the galvanometer may be conjugate conductors is, in this case,

F_ H_ (R^ A__J2__ C " A + \C "" A> P+Q + R ~

Now the resistance of the connector Q is as small as we can make it. If it were zero this equation would be reduced to

F __ ff C A '

and the ratio of the resistances of the conductors to be compared would be that of C to A, as in Wheatstone's Bridge in the ordinary form.

In the present case the value of Q is small compared with P or with R, so that if we assume the points F, V so that the ratio of R to C is nearly equal to that of P to A, the last term of the equation will vanish, and we shall have

F:ff::C:A.

The success of this method depends in some degree on the per- fection of the contact between the wires and the tested conductors at S, S', T' and T. In the following method, employed by Messrs. Matthiessen and Hockin*, this condition is dispensed with.

Pig. 36.

352.] The conductors to be tested are arranged in the manner

• Laboratory. Matthiessen and Hockin on Alloys.

352.] MATTHIESSEN AND HOCKIN's METHOD. 445

already described, with the connexions as well made as possible, and it is required to compare the resistance between the marks SS' on the first conductor with the resistance between the marks T Ton. the second.

Two conducting points or sharp edges are fixed in a piece of insulating material so that the distance between them can be accurately measured. This apparatus is laid on the conductor to be tested, and the points of contact with the conductor are then at a known distance SS'. Each of these contact pieces is connected with a mercury cup, into which one electrode of the galvanometer may be plunged.

The rest of the apparatus is arranged, as in Wheatstone's Bridge, with resistance coils or boxes A and C, and a wire PR with a sliding contact piece Q, to which the other electrode of the galva- nometer is connected.

Now let the galvanometer be connected to S and Q, and let Al and C1 be so arranged, and the position of Q so determined, that there is no current in the galvanometer wire.

Then we know that XS A

where XS, PQ, &c. stand for the resistances in these conductors. From this we get

XY '

Now let the electrode of the galvanometer be connected to S*, and let resistance be transferred from C to A (by carrying resistance coils from one side to the other) till electric equilibrium of the galvanometer wire can be obtained by placing Q at some point of the wire, say Q2. Let the values of C and A be now C2 and A2i and let A2 + C2 + PR = A± + CL + PR = R.

Then we have, as before, XS' _ XT' R

XY' R

In the same way, placing the apparatus on the second conductor at TT' and again transferring resistance, we get, when the electrode is in T\ YT'

XY " R

44:6 MEASUREMENT OF RESISTANCE. [S53»

and when it is in T,

XT

XT' R

Whence T'T _ A,-

XT'

We can now deduce the ratio of the resistances SS' and TfTt for

When great accuracy is not required we may dispense with the resistance coils A and C9 and we then find

88' T'T

The readings of the position of Q on a wire of a metre in length cannot be depended on to less than a tenth of a millimetre, and the resistance of the wire may vary considerably in different parts owing to inequality of temperature, friction, &c. Hence, when great accuracy is required, coils of considerable resistance are intro- duced at A and C, and the ratios of the resistances of these coils can be determined more accurately than the ratio of the resistances of the parts into which the wire is divided at Q.

It will be observed that in this method the accuracy of the determination depends in no degree on the perfection of the con- tacts at 8, y or T, T'.

This method may be called the differential method of using Wheatstone's Bridge, since it depends on the comparison of ob- servations separately made.

An essential condition of accuracy in this method is that the resistance of the connexions should continue the same during the course of the four observations required to complete the deter- mination. Hence the series of observations ought always to be repeated in order to detect any change in the resistances.

On the Comparison of Great Resistances.

353.] When the resistances to be measured are very great, the comparison of the potentials at different points of the system may be made by means of a delicate electrometer, such as the Quadrant Electrometer described in Art. 219.

If the conductors whose resistances are to be measured are placed in series, and the same current passed through them by means of a battery of great electromotive force, the difference of the potentials

355-] GREAT RESISTANCES. 447

at the extremities of each conductor will be proportional to the resistance of that conductor. Hence, by connecting the electrodes of the electrometer with the extremities, first of one conductor and then of the other, the ratio of their resistances may be de- termined.

This is the most direct method of determining resistances. It involves the use of an electrometer whose readings may be depended on, and we must also have some guarantee that the current remains constant during the experiment.

Four conductors of great resistance may also be arranged as in Wheatstone's Bridge, and the bridge itself may consist of the electrodes of an electrometer instead of those of a galvanometer. The advantage of this method is that no permanent current is required to produce the deviation of the electrometer, whereas the galvanometer cannot be deflected unless a current passes through the wire.

354.] When the resistance of a conductor is so great that the current which can be sent through it by any available electromotive force is too small to be directly measured by a galvanometer, a condenser may be used in order to accumulate the electricity for a certain time, and then, by discharging the condenser through a galvanometer, the quantity accumulated may be estimated. This is Messrs. Bright and Clark's method of testing the joints of submarine cables.

355.] But the simplest method of measuring the resistance of such a conductor is to charge a condenser of great capacity and to connect its two surfaces with the electrodes of an electrometer and also with the extremities of the conductor. If E is the dif- ference of potentials as shewn by the electrometer, S the capacity of the condenser, and Q the charge on either surface, R the resist- ance of the conductor and x the current in it, then, by the theory of condensers, Q = SE.

By Ohm's Law, E = Ex,

and by the definition of a current,

*--^. dt

Hence -Q=RS^,

t_

and Q = Q0ef*s,

where QQ is the charge at first when t = 0.

448

MEASUREMENT OF RESISTANCE.

[356.

Similarly E — EQ e RS'

where H0 is the original reading of the electrometer, and E the same after a time t. From this we find

which gives J2 in absolute measure. In this expression a knowledge of the value of the unit of the electrometer scale is not required.

If S, the capacity of the condenser, is given in electrostatic measure as a certain number of metres, then R is also given in electrostatic measure as the reciprocal of a velocity.

If S is given in electromagnetic measure its dimensions are

fZ

-j-, and R is a velocity.

Since the condenser itself is not a perfect insulator it is necessary to make two experiments. In the first we determine the resistance of the condenser itself, 720, and in the second, that of the condenser when the conductor is made to connect its surfaces. Let this be R'. Then the resistance, R, of the conductor is given by the equation

_L.. jL JL

R R' RQ

This method has been employed by MM. Siemens.

Thomson's * Method for the Determination of the Resistance of

the Galvanometer. 356.] An arrangement similar to Wheatstone's Bridge has been

Galvanometer

Fig. 37. employed with advantage by Sir W. Thomson in determining the

• Proc. R. 8., Jan. 19, 1871.

357-] MANCE'S METHOD. 449

resistance of the galvanometer when in actual use. It was sug- gested to Sir W. Thomson by Mance's Method. See Art. 357.

Let the battery be placed, as before, between B and C in the figure of Article 347, but let the galvanometer be placed in CA instead of in OA. If Ifi—cy is zero, then the conductor OA is conjugate to BC, and, as there is no current produced in OA by the battery in BC, the strength of the current in any other conductor is independent of the resistance in OA. Hence, if the galvano- meter is placed in CA its deflexion will remain the same whether the resistance of OA is small or great. We therefore observe whether the deflexion of the galvanometer remains the same when 0 and A are joined by a conductor of small resistance, as when this connexion is broken, and if, by properly adjusting the re- sistances of the conductors, we obtain this result, we know that the resistance of the galvanometer is

•-3F-

where c, y, and /3 are resistance coils of known resistance.

It will be observed that though this is not a null method, in the sense of there being no current in the galvanometer, it is so in the sense of the fact observed being the negative one, that the deflexion of the galvanometer is not changed when a certain con- tact is made. An observation of this kind is of greater value than an observation of the equality of two different deflexions of the same galvanometer, for in the latter case there is time for alteration in the strength of the battery or the sensitiveness of the galvanometer, whereas when the deflexion remains constant, in spite of certain changes which we can repeat at pleasure, we are sure that the current is quite independent of these changes.

The determination of the resistance of the coil of a galvanometer can easily be effected in the ordinary way of using Wheatstone's Bridge by placing another galvanometer in OA. By the method now described the galvanometer itself is employed to measure its own resistance.

Mance's* Method of determining the Eesistance of the Battery.

357.] The measurement of the resistance of a battery when in action is of a much higher order of difficulty, since the resistance of the battery is found to change considerably for some time after

• Proc. R. S., Jan. 19, 1871. VOL. I. G g

450 MEASUREMENT OF RESISTANCE. [357.

the strength of the current through it is changed. In many of the methods commonly used to measure the resistance of a battery such alterations of the strength of the current through it occur in the course of the operations, and therefore the results are rendered doubtful.

In Mance's method, which is free from this objection, the battery is placed in BC and the galvanometer in CA. The connexion between 0 and B is then alternately made and broken.

Now the deflexion of the galvanometer needle will remain un- altered, however the resistance in OB be changed, provided that OB and AC are conjugate. This may be regarded as a particular case of the result proved in Art. 347, or may be seen directly on the elimination of z and ft from the equations of that article, viz. we then have

If y is independent of #, and therefore of /3, we must have a a = cy. The resistance of the battery is thus obtained in terms of c, y, a.

When the condition a a = cy is fulfilled, the current through the galvanometer is then

Ea Ey

-> or

ab--y(a +

To test the sensibility of the method let us suppose that the condition cy = a a is nearly, but not accurately, fulfilled, and that

Fig. 38.

y0 is the current through the galvanometer when 0 and B are connected by a conductor of no sensible resistance, and y^ the current when 0 and B are completely disconnected.

To find these values we must make /3 equal to 0 and to oo in the general formula for y, and compare the results.

357-] COMPARISON OF ELECTROMOTIVE FORCES. 451

The general value for y is

where D denotes the same expression as in Art. 348. Making use of the values of y given above we can then easily shew that the expressions for yQ and yl are approximately

c(cy-aa) y*

and -

y(y+a) From these values we find

The resistance, c, of the conductor AB should be equal to a, that of the battery; a and y should be equal and as small as possible; and b should be equal to q + y.

Since a galvanometer is most sensitive when its deflexion is small, we should bring the needle nearly to zero by means of fixed magnets before making contact between 0 and B.

In this method of measuring the resistance of the battery, the current in the galvanometer is not in any way interfered with during the operation, so that we may ascertain the resistance of the battery for any given strength of current in the galvanometer so as to determine how the strength of the current affects the resistance *.

If y is the current in the galvanometer, the actual current through the battery is XQ with the key down and ^ with the key up, where

/ b ac \ / b \

y

the resistance of the battery is

cy

a = a

and the electromotive force of the battery is

C ,„ X \