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And by adding unity to each side we have, from equations (1) and (2)

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=

. (4) (5)

a+b=whole length of bridge wire a'+b'.

.. R+x+ao+s+y+bo=s+x+a'o+R+Y+b'o

Hence from (3)

S+Y+bσ=R+Y+b'o;

.. R−s = (b − b') σ = (a' − a) σ, by (4).

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Now (aa) is the resistance of a portion of the bridge wire equal in length to the distance through which the sliding-piece has been moved. This distance can be measured with very great accuracy, and thus the difference of the resistances of the two coils can be very exactly determined.

To obtain all the accuracy of which the method is capable, it is necessary that the contacts should be good, and should remain in the same condition throughout. Mercury cups should generally be employed to make contact, and it is necessary that the electrodes of the various coils should be pressed firmly on to the bottoms of these either by weights, or, if convenient, by means of spring clamps.

At the three points C, N, N', we have contacts of two dissimilar metals. These points are probably at different temperatures-the observer's hand at c tends to raise its temperature—and a difference of temperature in a circuit of different metals will, it is known, produce a thermoelectric current in the circuit. This current will, under the circumstances of the experiment, be very small; still, it may be a source of error.

The best method of getting rid of its effects is to place a commutator in the battery circuit, and make two observations of each of the lengths a and a', reversing the battery between the two. It can be shewn that the mean of the two observations gives a value free from the error produced by the thermo-electric effect.

Again, a variation in the temperature of a conductor produces an alteration in its resistance. For very accurate work it is necessary to keep the coils R and s at known temperatures. This is generally done by means of a water-bath, in which the coils are immersed.

It has been found that for most of the metals, at any rate within ordinary limits of temperature, the change of resistance per degree of temperature is very nearly constant, so that if R be the resistance of a coil at temperature ° C., R。 its resistance at o°, and a the coefficient of increase of resistance per degree of temperature, we have

R = R。 (I+at).

Carey Foster's method is admirably adapted for finding this quantity a. The standard coil s is kept at one definite temperature, and the values of the difference between its resistance and that of the other coil are observed for two tem

peratures of the latter. Let these temperatures be t, and t1⁄2, and the corresponding resistances R, and R; then we have

a = (R-R2) /R。 (t1-t2).

The observations have given us the values of R1-s and R1 -s with great accuracy, and from them we can get R1-R2; an approximate value of R. will be all that is required for our purpose, for it will be found that a is a very small quantity, and we have seen (p. 44) that we may without serious error employ an approximate value in the denominator of a small fraction.

Whenever precautions are requisite to maintain the coils at a uniform temperature, the interchanging of the

coils R, S is a source of difficulty with the ordinary arrange ments. Time is lost in moving the water-jackets in which the coils are immersed, and the temperature may vary. The contacts, moreover, are troublesome to adjust. To obviate this, among other difficulties, a special form of bridge was. devised by Dr. J. A. Fleming, and described in the 'Pro ceedings of the Physical Society of London,' vol. iii. The ordinary bridge may be easily adapted to an arrangement similar to Fleming's, as follows. EGFH (fig. 74) are four mercury cups; E and F

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FIG. 74.

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spring clamps, while the electrodes of s are in G and H.

For the second observation the electrodes of R are placed in G and H, those of s in E and F, as shewn by the dotted lines. This interchange is easily effected. The water jackets need not be displaced; the coils can readily be moved in them.

The connections A E, MF, &c., may conveniently be made of stout copper rod, fastened down to a board of dry wood, coated with paraffin. To make the mercury cups. the ends of these rods are turned up through a right angle. and cut off level. They are then amalgamated and short pieces of india-rubber tubing are slipped over them, and tied round with thin wire; the india-rubber tubing projects above the rod, and thus forms the cup. The other ends of the rods are made to fit the binding screws of the ordinary bridge.1

For a fuller account of this and other similar contrivances, see Philosophical Magazine, May 1884.

Calibration of a Bridge-wire.

The method gives us also the best means of calibrating a bridge-wire. Make an observation exactly as above Alter the value of P slightly by inserting in series with it a short piece of German-silver wire. The only effect will be to shift somewhat the positions of c and c' along the scale, and thus the difference between R and s is obtained in terms of the length of a different part of the bridge-wire. If the wire be of uniform section the two lengths thus obtained will be the same. If they are not the same, it follows that the area of the cross-section, or the specific. resistance of the wire, is different at different points, and a table of corrections can be formed as for a thermometer (p. 242).

If the difference between the two coils be accurately known we can determine from the observations the value of the resistance of a centimetre of the bridge-wire. This is given by equation (6); for the values of R-S and a'-a are known, and we have

σ= · (R −S) | (a' — a).

For this purpose the following method is often convenient. Take two 1-ohm coils and place in multiple arc with one of them a 10-ohm coil. Let the equivalent resistance of this combination be R; then the value of R is 10/11 ohms. Instead of interchanging the coils place the ten in multiple arc with the other single ohm and make the observation as before; then in this case we have

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and if be the distance through which the jockey has been moved we obtain

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Experiments.

(1) Calibrate the bridge-wire.

(2) Determine the average resistance of one centimetre of it. (3) Determine accurately the difference between the resistance of the given coil and the standard 1-ohm at the temperature of the room.

Enter results thus :

(1) Value of R-s for calibration, '009901-being the difference between 1 ohm and 1 ohm with 100 in multiple arc

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(2) R-S- *09091 ohm. 7 (mean of 5 observations) = 50.51 cm.

σ = '00179 ohm.

(3) Difference between the given coil and the standard at temperature of 15° C., observed three times.

Values 0037, 0036, 00372 ohm. Mean 00367 ohm.

80. Poggendorff's Method for the Comparison of Electromotive Forces. Latimer-Clark's Potentiometer.

The method given in § 76 for the comparison of electromotive forces is subject to a defect similar to that mentioned in §77, on the measurement of resistance; that is, it depends upon measuring the deflexion of a galvanometer needle, and assumes that the E. M.F. of the batteries employed remain constant throughout the experiment.

The following method, first suggested by Poggendorff, resembles the Wheatstone-bridge method for measuring resistances, in being a null method; it depends, that is to say, on determining when no current passes through a galvanometer, not on measuring the deflexion.

We have seen

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