ELECT 11 ICITY [ELECTRIC CURRENT. the deflection 1 due to a current c in one is me, and in the other nc, where m and n are the " constants" of the two coils, then the deflection for currents c l and c 2 is mc^ nc 2 . Fig. 2 1 gives a scheme of the arrangement for measuring resistances with this instrument. V is the battery in serted in the com mon branch ED of the two circuits, which convey cur rents dividing off at D, and going in opposite direc tions round the coils of G. If we wish to measure the resistance of a wire, it is inserted at AB by means of binding screws or mercury cups, and the resistance of the other circuit is varied until there is no deflection ; then AB is replaced by a known resistance, which is made up until there is zero deflection as before. It is obvious that the only requisite here is that the resistances of EFK, EA, BL, and the galvanometer coils should remain constant. Variations in the electromotive force or internal resistance of the battery dc not affect the result. The method which we have thus sketched is the best way of nsing the differential galvanometer, and it does not matter even if tho coils are not exactly symmetrical. Let the constants of the coils M and N be m and u, so that the deflection due to currents c t and c 2 in M and N is m^-nc^ . Let the resistance from E to U in the single branch be B, and in the circuits EFK and EABL, which pass round M and N respectively, R and S + U , U being the resistance between A and B , which is such that the deflection is zero. Then ( ) E = wcj - nc z = j m(S + U) - wR j ^ ... (n), where E is the electromotive force of the battery, and D = (R + S + U)B + R(S + U) . Suppose we substitute U for U, and arrange U so that we have- again zero deflection. Then
m(S From a and we get U = U . For farther details concerning this method, see Maxwell, vol. i. 346, and Schwendler, Phil. Mag., 1867. The differential galvanometer method was much used by Becquerel and others, but it is now entirely superseded as a practical method in this country by the Wheatstone s bridge method. Suppose we have a circuit ABDC of four conductors. Insert a galvano meter G between B and C, and a battery between A and D. Adjust say the resistance AB until the galvanometer in BC indicates no current. The bridge is then said to be balanced, and the potentials at B and C must be equal. But the whole fall of potential from A to D along ABD is the same as that along ACD ; hence if the fall from A to B is to be equal to that from A to C, we must have R_T S~U where R,S,T,U are the resistances in AB, BD, CA, DC. This is the condition that BC and AD be conjugate. We might have deduced it as a particular case of the general theory given above. Hence if we know the resistances S,T,U, we Fig. 22. 3 The deflections are supposed small. ST get in terms of these R = . S is often called the standard resistance, and T, U the arms of the bridge or balance. The sensibility of this arrangement may be found practi cally by increasing or decreasing R so as to derange the balance. The largest increase which we can introduce without producing an observable galvanometer deflexion measures the sensibility of the bridge. If we had a given set of four conductors, and a battery Arrang and galvanometer of given resistance, then it may be men * fo shown (see Maxwell, vol. i. 348) that the best arrange- meiit is that in which the battery or galvanometer connects the junction of the two greatest resistances with that of the two least, according as the former or the latter has the greater resistance. The practical problem might take another form. We might have given a resistance, and have at our disposal known resistances of any desired magnitude to form our bridge. We might also suppose further that we had given the total area of the plates of our battery, and the dimensions of the channel in which the galvano meter wire was to be wound. We may neglect the thick ness of the silk coating, or assume that it is proportional to the thickness of the wire. Then, B and G being the resistances of the battery and galvano meter, the electromotive force E a V B, and the number of turns in the galvanometer aV(5. Let us put S = ?/R, T = ~R, and TJ = i/zR. These resistances would balance ; let us however put (l+a;)R in the bra"nch AB in stead of R, the others being unchanged, and calculate the effect on the galvanometer in G, which we put proportional to the current in BO, and to the number of turns on galvanometer. Then, from equation (77) (or Maxwell, vol. i. 349), we find that the deflection 5 varies as __ (l + 2/)(l+2)BG + y(l + in order that 8 may be a maximum, we must have (a). (/B), (7), (8). a and /3 give at once by addition and subtraction B 2(1 + ?/) 2 -5 and BQ =yzli 2 > (0- Combining the four equations (7), (8), (e), (f), we It appears, therefore, that whsn all the resistances on the bridge are at our disposal, we ought to make them all equal to the resistance to be measured, or come as near this as we can ; e.g., if we had a very small resistance to measure, we should make the arms of the bridge small, and take a small-resistance in preference to a high-resistance galvanometer. In order to carry out measurements of resistance with stand- ease we must possess a series of graduated resistances, with ards f which we can compare any unknown resistance, and of re which we can make the arms of our balance, A r c. Again, if the measurements of one electrician are to be of any use to another, there must be a common standard. It would be most convenient to have only one standard for all nations, and this standard might be either arbitrary, like the standard of length, or absolute in some sense such as we have defined above. Arbitrary standards have at different times been proposed by Jacobi and others. The mercury standard of Siemens, to which we alluded in the historical sketch, has obtained great prevalence on the Continent.
The British Association unit or ohm is an absolute unit,
