(r+r) bear to (s + s'), there will be on the branch A B' C points of the same potentials as those of any points on ABC respectively. Now if we connect two points B and B' by a 'bridge' BG B' containing a galvanometer G, then according as we see indicated a current from B to B', no current, or a current from B' to B, we shall conclude that B is above B' in potential, B and B' are at the same potential, or B' is above B in potential, respectively. The case of no current through G is the most important; in this case B and B' must be at the same potential. Now in fig. ii. M P represents the potential VA; and we have taken Vc as our zero potential, so that the point C is represented by O or by O'. The fall of potential down the one circuit is given by the line PQO, the fall down the other circuit by PQ'O. The points Q and Q' represent B and B' respectively; when these latter are at the same potential, as proved by the absence of current in G. The rest of the figure is clear after what we have said in Chapter XIII. § 7. Then when B and B' are at the same potential, or when QNQ'N', we have, by Euclid, Book VI. Hence one of the resistances is determined by the other three. Algebraic proof of the same.-We will now use a notation that will be readily understood with reference to fig. i. of § 4, the notation being similar to that employed in Chapter XIII. §§ 9 and 10. Since there is no current through G, we have C, Cr.r- Cs. s = 0; or C. r = = Cy; and C, Cs'. BG B'A, we have Cs. S. Applying the same law to the circuit BC B'G B, and remembering that Cr = Cr, and C1 = Cs, we have also in like manner Cr. r - Cs. s' = 0; or C. r = Cs. s'. Whence we readily obtain the result that It is clear that this formula remains true, however the E.M.F. of the battery may alter. In this fact lies the main superiority of this method, over the previous methods, of measuring resistance. $5. Slide-form of Wheatstone's Bridge.-The figure here given represents roughly the simplest and least expensive form of Wheatstone's bridge. On a board are fixed three bands, A, B, and C, of stout copper; the pieces A and C being elbow-shaped. Between A and B, and B and C, are gaps; and here can be inserted, most conveniently by means of mercury cups, any resistances we please. The other ends of the pieces A and C are joined by a thin uniform wire of high specific resistance; and parallel to this runs a scale that should G have its initial and final points exactly opposite the points where the wire is soldered to the thick copper pieces. This scale is graduated to read both ways; and, on the supposition that the wire is of uniform resistance, and that the scale corresponds properly to the beginning and end of the wire, we can read off the ratio of the two resistances s and s' into which any point such as B' divides the wire. Over the wire slides a spring key B'; by pressing the button of this key we make contact with the wire by means of a metallic edge or fine wire. The point where contact is thus made should be exactly opposite to the point on the scale at which the little index carried by the key is then pointing. The 'bridge' B G B' connects the piece B and the key B'; the battery terminals are connected with A and C. Referring to fig. i. of § 4, we see that we have complete correspondence; only the points A, B, and C of that figure have been replaced by the metal pieces A, B, and C, which are, from a resist ance point of view, equivalent to points. Further, we see that the point B' is moveable, so that we can make the ratio s:s anything that we please. In using the instrument we insert our unknown resistance at r, and a resistance box at r'. We then adjust the position of the key B' until the making of contact with the wire at B' gives no current through the galvanometer in the bridge. When this is the case, B and B' must be at the same potential; and we then have. which determines r. r = S Note.-There is often some want of exact correspondence between the ends of the scale and the ends of the wire. Such a defect produces less error in the result if we have s and s not very unequal. We can contrive this by having a resistance box at r', and by removing the plugs (i.e. by throwing in resistance) until we get our zero deflexion of G when B' is not far from the centre of the wire; in other words we make not very unequal to r. § 6. Wheatstone's Bridge; Resistance Box Form.-The above described instrument is liable to error. The wire, by much use, loses its uniformity; so that equal lengths no longer correspond to equal resistances. Moreover, there is, even in the best instruments, some uncertainty as to the exact correspondence between the point of contact B' and the point on the scale against which the index rests. (This second defect can, however, be eliminated.) In the instrument used for technical purposes B' is fixed, and s and s' are resistance boxes containing coils of (e.g.) I, 10, 100, and 1,000 ohms. For the known resistance we have another resistance box. We can thus, by means of the two boxes s and s', measure from th to 1,000 times any resistance in the box r'; I TOOO this is simply a matter of removing the suitable plugs. We can thus with certainty measure to th of 1 ohm, which is better I 1000 than measuring to smaller fractions with uncertainty. § 7. Resistance of a Galvanometer. This can be measured as any other resistance; but a very simple method is as follows. Sir Wm. Thomson's method.-Here we put the galvanometer G, whose resistance is required, in the place of one of the resistances ; and, in the place of the bridge with its galvanometer, we have a key K, by means of which we can make or break connection between B and B' at will. The galvanometer G will have a deflexion due to the current through AB; the battery power and resistance in the external circuit A P C must be so adjusted as to give a reasonable deflexion of G. B IK We adjust the resistances s and s' until the making of contact with the key K produces no alteration in the deflexion of G. When this is the case, it can be proved that no current then flows through the bridge; and that as before, G = . ', where G is the galvanometer resistance. § 8. Resistance of a Battery-Cell. The measurement of the resistance of a battery-cell is not at all a simple matter. For whereas in the above measurements we had only one constant unknown, viz. the constant resistance x that we desired to measure, in the case of the battery-cell we have two unknowns, one of which is variable from moment to moment; we have, in fact, the resistance x, and also the E.M.F. y, which, owing to varying degrees of polarisation, is not constant. We here give two methods for measuring x. I. The E.M.F. reduced to zero.—If we have any even number 2 m of similar cells, and if we set m against m, the total E.M.F. will be zero. When the current, employed in measuring the resistance, passes through these 2 m cells, the polarisation produced should still be zero in total amount. With cells thus matched m against m, we can measure the resistance of the whole 2 m cells as a 'dead' resistance; that of one cell is found by dividing by 2 m. Against this method it may be stated that, as a rule, we require the resistance of the cell or battery as it is at any given time ; and we cannot generally provide another cell or battery, equal to this in E. M.F., to oppose to it. We may call this method the 'Method of opposition.' II. Mance's method.-Let us arrange a Wheatstone's bridge as in § 5, but with the cell or battery, whose resistance is to be determined, in the place of It can be shown that if the resistances be so adjusted that G maintains the same deflexion whether the key K be opened of closed, then we have the old relation. The proof is not very simple. By application of Kirchhoff's laws we obtain equations by means of which the unknowns can be determined. The current through G will be found to involve the external resistance of A K C in such a way that it is only independent Whence it follows that, if this current is the of it when r r' = S same whether the key is open (giving an infinite resistance) or closed (giving a finite resistance), this relation must hold. (For further discussion of this method see 'Phil. Mag.,' Series V., Vol. 3, 1877, Professor O. J. Lodge; and also in Glazebrook and Shaw's 'Practical Physics.') § 9. Measurement of E.M.F.-Let us take our standard cell, whose E.M.F. we suppose (for simplicity) to be just 1 volt, and let us measure the AV between its terminals when the circuit is |