activity wasted in heating the conductors in circuit is (E- E1)C. This ratio E1/Eis the same as that found above in the case of a generator and a motor, and may be called as before the electrical efficiency of the arrangement.

Hence, in order that as nearly as possible the whole electrical energy given out in the circuit may be spent in charging the battery, as many cells should be placed in circuit as suffice to nearly balance the electromotive force E of the generator, that is, the charging should be made to proceed as slowly as possible. In practice, however, a very slow rate of charging is not economical, as the work spent in driving the generator, if a dynamo- or magnetoelectric machine, against frictional resistances would be greater than the useful work done in the circuit; and if the speed of the generator slackened for a little the battery would tend to discharge through it.

As in the case of the motor (p. 265), the electrical efficiency of the arrangement can be increased by increasing E and E1, so that E E is maintained constant. E may, in the present case, be increased by running the generator faster, or by using a machine adapted to give higher potentials. As before, if E be increased to n E, while E1 is changed to E, so that nE – E'1 = E — E1, the electrical efficiency becomes (n - 1)/n + E1/nE as in (7) above.

1

The electromotive force of a Faure or storage cell is about 2.2 volts when fully charged, but is considerably less when nearly discharged. When the cell is placed in the charging circuit, the counter electromotive force which it opposes rises quickly to a little less than this value, and thereafter gradually increases, while the charging current falls in strength. In order to measure, therefore, the whole energy spent in charging a secondary battery, we must either use some form of integrating energy-meter which

gives accurate results, or measure, at short intervals of time, V with a potential galvanometer, and C with a current galvanometer placed permanently in the circuit. After the battery has been charged, the total number of joules spent is obtained by multiplying each value of VC by the number of seconds between the instant at which the corresponding readings were taken and that at which the next pair of readings were taken, and adding all the results. Or, more exactly, values V and C are obtained for each interval by finding the arithmetic means of the values of V and of C at the beginning and end of each interval, and taking the product of these two means as the value of the activity for that interval. Each product is multiplied by the number of seconds in the corresponding interval, and the sum of the products is the whole energy spent. The integral work in joules having been thus estimated, the efficiency of the battery may be obtained by finding in the same manner the total number of joules given out in the external working circuit while the battery is discharging. The ratio of the useful work thus obtained to the whole work spent in charging is the efficiency of the battery. In discharging in an electric light circuit, the greatest economy is obtained when the resistance of the working part of the circuit is very great in comparison with that of the battery and main conductors. Neglecting the latter part of the resistance, we see that, if a large number of lamps are arranged in multiple arc, a large number of cells should also be joined in multiple arc, so that, while the requisite difference of potential is obtained the resistance of the battery is still small in comparison with that of the external circuit.

As regards the measurement of energy spent in electric light circuits, in which continuous currents are flowing,

we have already sufficiently indicated above (Chap. VII.) how this may be done. To find the activity, or work spent per unit of time, in any part of a circuit, we have only to find the dffference of potential, V, in volts between its extremities with a potential galvanometer, and the current, C, in amperes flowing through it with a current instrument. If the activity be constant, we have simply to multiply VC by the number of seconds in any interval of time, to find the number of joules spent in that time in the part of the circuit in question. If the activity is variable, the whole energy spent in any time may be estimated by finding VC at short intervals of time, and calculating the integral as explained above (p. 272).

So far we have been considering only measurements made in the circuits of batteries or of continuous current generators. Alternating machines in which the direction of the current is reversed two or three hundred times a second are, however, frequently employed, especially in electric light circuits, and it is necessary therefore to consider the methods of electrical measurement available in such cases. We shall consider briefly, first, a class of instruments some of which can be used in a circuit of either kind, and we shall deal in the first place with their application in continuous current circuits.

These are instruments the fundamental principle of which is the mutual electromagnetic action between two circuits, to be calculated conveniently in most cases in which this can be done, by replacing each circuit according to Ampère's theory, by an equivalent magnetic shell (p. 42), and considering the mutual action of the systems. Sir William Thomson's Electric Balances, described in Chap. VII.,are instruments which act on this principle, and can

T

be used both in alternating and in continuous current circuits.

Weber's well-known electrodynamometer* is another instrument of this class, but arranged to be used as an absolute or standard current-meter. It may be considered as constructed by replacing the needle of the standard tangent galvanometer (Chap. IV.) with a coil of radius small in comparison with that of the galvanometer coil, and suspended by a torsion wire or bifilar, so as to hang in equilibrium when no current is passing through it, with its plane at right angles to that of the large coil. Hence when a current C is made to flow through the fixed coil, and a current C' through the suspended coil, a couple is exerted on the latter, tending to set it with its plane parallel to the large coil, and this tendency is resisted by the action of the torsion or bifilar suspension, so that there is equilibrium for a deflection 0, the magnitude of which plainly depends on the product CC' of the two current strengths. To avoid disturbance from the action of the local horizontal magnetic force, the large coil may be placed parallel to the direction of that force, and the small coil brought back when deflected by the current to the initial position at right angles to the large coil, by turning the upper end of the suspension wire or wires through a measured angle.

By Ampère's theorem (p. 42) the suspended coil is equivalent to a small needle of moment n A C', where n is the number of turns of wire in the coil, A their mean area. Hence if N be the number of turns in the large coil, its mean radius, we have as in p. 47 for the electromagnetic couple on the suspended coil, when brought back to the * See Maxwell's Electricity and Magnetism, vol. ii. p. 330.

initial position, the value 27/r. NC.nA C'or 2m/r.nNACC This couple is balanced by the opposite couple given by the supension, the magnitude of which for all angles within a certain range is supposed known from experiment. Calling this latter couple L, we have

If the two coils be joined in series so that the same current flows through both, we have C = C', and therefore

With such an instrument therefore an absolute measurement of a current can be made without its being necessary first to determine H.* In practical work the instruments on this principle usually employed are such as require to have their constants determined by comparison with standard instruments, such as a standard tangent galvanometer, or a standard dynamometer, and are dealt with in Chap. VII. We may here mention, however, Siemens' electro-dynamometer, in which a suspended coil is acted on by a fixed coil, and the strength of the current deduced, by means of a table of values for different angles, from the torsion which must be given to a spiral spring to bring the coil back to the zero position.

When an instrument on this principle is arranged for use as an activity-meter, one set of coils, the fixed or the movable, is made of thick wire so as to carry the whole current in the circuit, while the other set is made of high resistance and is connected to the two ends of the part of

For fuller particulars regarding absolute instruments of this class see Theory and Practice of Absolute Measurements in Electricity and Magnetism, vol. ii.