To ascertain the degree of accuracy of this estimate, let the altered quanti- ties in the second observation be accented, then D δ, E D′ m(B + β) − n(A′ + α) = ′ δ ′ . E m(B + β) − n(A + α) = Hence n(A′ − A) = D D′ δ − ′ δ′. E E If δ and δ ′ , instead of being both apparently zero, had been only observed to be equal, then, unless we also could assert that E = E ′ , 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 δ and δ ′ 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 ascertained by tak- ing 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 B, and then observing the resulting deviation of the galvanometer needle, we can estimate the number of degrees corresponding to an error of 1%.

To find the actual degree of precision we must estimate theMEASUREMENT OF RESISTANCE.

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 B, and if the positions of A and B are exchanged, then the second equation becomes m(A + β) − n(B + α) = whence (m + n)(B − A) = D′ ′ δ, E D D′ ′ δ− δ. E E If m and n, A and B, α and β are approximately equal, then B−A= 1 (A + α)(A + α + 2r)(δ − δ ′ ). 2nE Here δ − δ ′ may be taken to be the smallest observable deflexion of the gal- vanometer. 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 α as the square (A + α)(A + α + 2r) of the length. Hence there will be a minimum value of n when 3 r2 α = 13 (A + r) 2 1 − −1 . 4 (A + r)2 { √ } If we suppose r, the battery resistance, small compared with A, this gives α = 13 A; or, the resistance of each coil of the galvanometer should be one-third of the resistance to be measured. We then find 8 A2 B−A= (δ − δ ′ ). 9 nE ∗ This investigation is taken from Weber’s treatise on Galvanometry. Göttingen Transactions, x. p. 65.

DIFFERENTIAL GALVANOMETER.

If we allow the current to flow through one only of the coils of the gal- vanometer, and if the deflexion thereby produced is Δ (supposing the deflex- ion strictly proportional to the deflecting force), then Δ= Hence mE 3 nE 1

if r = 0 and α = A. A+α+r 4 A 3 B − A 2 δ − δ′

. A 3 Δ

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 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 gal- vanometer 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.

221*.] Wheatstone’s Bridge consists essentially of six conductors connect- ing four points. An electromotive force E is made to act between two of the points by means of a voltaic battery introduced between B and C. The current between the other two points O 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.208