ratio of this to E C is CR/E. But CR is equal to the constant difference E-E1, hence the ratio is (E – E1)/E, and this becomes smaller as E is increased. A greater efficiency is therefore obtained by using high potentials than by using low potentials. Hence a greater electrical efficiency is realized, with a given magneto- or dynamoelectric machine used as generator and a given motor, when both generator and motor are run at higher speeds. Consequently the generator should be run as fast as possible, and the motor loaded lightly, or the speed with which the working resistance is overcome reduced by gearing between it and the motor.

When high potentials are obtained by the use of machines wound with fine wire, or by using as generator a battery of a large number of cells joined in series to drive a high potential motor, the gain of electromotive force is accompanied by an increase of resistance in the circuit. But if we suppose the speed of the motor to be so regulated that the difference between the total electromotive force in the circuit and the back electromotive force of the motor remains the same in the different cases, it is easy to show that the electrical efficiency of the arrangement is greater for high electromotive forces than for low. If, as supposed, E-E1 remains constant, while E is changed to n E, we have for the total activity of the motor nEC (E – E1) C. Dividing this by nEC we get for the electrical efficiency,

As n is made greater and greater, the first term on the right becomes more and more nearly equal to unity, and the last term to zero. Hence, on the supposition made,

the efficiency is increased by increasing the working electromotive forces. Taking as a particular case n=2, we see that the efficiency is together with one-half of the former efficiency; if n = 4, the efficiency is together with one-fourth of the former efficiency, and so on for other values of n. This result holds for any case whatever in which the condition that E-E1 should remain constant is fulfilled; and hence it is independent of any change that may have been made in the resistances of the generator or motor in order to obtain the higher electromotive force n E. For example, it is plain that no sensible change in the actual rate of loss by heating of the conductors by the current will be produced by increasing the resistances of the generator and motor, if these be very small in comparison with the remainder of the resistance in circuit; as, since E-E1 remains constant and the resistance is practically the same as before, the current strength will not be perceptibly altered. The ratio, however, of the activity wasted in heating to the total activity will be only 1/nth of what it was before. In the opposite extreme case, in which the generator and motor have practically all the resistance in circuit, the current, C, (= (E — E1)/R} is diminished in the ratio in which the resistance is increased; and the actual rate of loss by heat according to Joule's law, (E- E1)2/R, is diminished in the same ratio, so that, as in the former case, its ratio to the total activity nEC is 1/nth of what it was for the electromotive force E. We see, therefore, that here also the efficiency must be the same in both cases.

We have called EE the electrical efficiency of the arrangement, but this is not to be confounded with the efficiency of the motor itself. The activity E1 C includes the wasted activity or rate at which work is done against

frictional resistances in the motor itself, and in the gearing which connects it with its load, as well as the useful activity or rate at which it performs useful work. Hence, although the electrical efficiency of the arrangement be very great, only a small amount comparatively of the energy given to the motor may be usefully expended, and vice versâ; and we define therefore the efficiency of a motor at any given speed as the ratio of the useful activity to the whole activity, taking as the latter the total rate at which electrical energy is expended in the motor; that is, EC+ CR1, or, which is the same, V C, where V is the difference of potential between the terminals of the motor. Accordingly, if A be the useful activity, we have for the efficiency of the motor the ratio A/VC.

To determine this ratio in any particular case the motor is run at the required speed, V is measured with a potential galvanometer, and C with a current galvanometer, and their product taken, or VC is determined with some form of electrical activity-meter, while A is determined by means of a suitable ergometer. A very convenient and accurate friction ergometer may be formed by passing a cord once completely round the pulley of the motor, and hanging a weight on the downward end, while the other is made to pull on a spiral spring fixed at its upper end and provided with an index to show its extension. The weight is adjusted so that the motor runs at the required speed, while wasting all its work in overcoming the friction of the cord, and the extension of the spring is noted, and the corresponding pull found in the same units of force as those used in estimating the downward pull due to the weight. Let the weight used in any experiment be taken in grammes, and be denoted by w, and let w' be the number of grammes required to

stretch the spring by gravity to the same amount, then the total force overcome is in dynes (w - w') g, where g is the acceleration, in centimetres per second per second, produced by gravity at the place of experiment (at London g = 981 71 nearly). If n be the number of revolutions per second, and c the circumference in cms. of the pulley at the part touched by the rope, the velocity with which this force is overcome is n c, and therefore the activity in ergs per second is n c (w reckoned in watts, we have the equation,

I

w') g. If A is

107

If w-w' be taken in pounds, and c in feet, and ʼn be the number of revolutions per minute, the activity in horse-power is given by

We have now considered cases in which electrical energy is transformed into mechanical work by means of motors working by electromagnetic action, and have seen that the whole electrical activity EC in the circuit is equal to the useful activity of the motor together with the unavailable part spent in heating the conductors in circuit, and in overcoming the frictional resistances opposing the motion of the motor. Part of the electrical energy developed by a generator may however be spent in affecting chemical decompositions in electrolytic cells placed in the circuit, as, for example, in charging a secondary battery or "accumulator." Each cell in which electrolytic action

takes place, so that the result is chemical separation at the plates of the constituents of the solution acted on, opposes a counter electromotive force to that causing the current, and the work done per second in each cell over and above that spent in heat according to Joule's law (p. 75), is equal to the product of this counter electromotive force into the strength of the current. In most cases the counter electromotive force exceeds the electromotive force required to effect the chemical decompositions, and the energy corresponding to the difference of electromotive force appears in the form of what has been called local heat in the electrolytic cells.

In the case of a secondary battery charged by the current from an electrical generator, which is the only case we shall here consider, the activity spent in the battery while it is being charged is equal to the product of the difference of potential existing between the terminals of the battery while the current is flowing, multiplied by the strength of the current. Let be this difference of potential in volts, and C the current strength in amperes, then VC joules is the whole work per unit of time spent in the battery. The whole activity spent in the circuit is EC, or VC + CR, where E is the total electromotive force of the generator, and R is the resistance of the generator and the wires connecting it with the secondary. Again if E1 volts be the electromotive force of the secondary battery, which may be measured by removing the charging battery for an instant and applying a potential galvanometer to the terminals of the secondary, the activity actually spent in charging the battery may be taken as EC. Hence the ratio of the activity spent in charging the battery to the whole activity in the circuit is E1(V+RC) or E1/E, and the