It can, in fact, work more efficiently if it be not expected to do its work so quickly. Siemens has, in fact, proved that if the motor be arranged so as to do its work at less than the maximum rate, by being geared so as to do much less work per revolution, but yet so as to run at a higher speed, it will be more efficient; that is to say, though it does less work, there will also be still less electric energy expended, and the ratio of the useful work done to the energy expended will be nearer unity than before.

The algebraic reasoning is as follows: -If E be the electromotive force of the generator when the motor is at rest, and c be the current which flows at any time, the electric energy W, expended in unit time, will be (as expressed in watts) given by the equation,

W = Ec = e ( E - e ) / R . (1)

When the motor is running, part of this electric energy is being spent in doing Work, and the remainder is wasting itself in heating the wires of the circuit. We have already used the symbol w for the useful work (per second) done by the motor. All the energy which is not thus utilized is wasted in heating the resistances. Let H represent this heat. Its mechanical value will be HJ, where J stands for Joule's equivalent. Then we shall have W = w + HJ. But by Joule's law the heat-waste of the current, whose strength is c, running through resistance R, is expressed by the equation:

HJ = C2R. Substituting this value above, we get:

W = w + c2R, ... (2) which we may also write w = W- c2R. But by equation (1) W = Ec, whence w Ec-c2R, ... (3) and writing for c its value, E-e/R, we get w= (E-e) ( E- { E-e})/R or,

E- e w = e --- • • . (4)

R.

Comparing equation (5) with equation (1), we get the following:w/W=e(E-e) / E ( E-e); or, finally, w/W = e/E.

This is, in fact, the mathematical law of efficiency, so long misunderstood until Siemens showed its significance. It may appropriately be called the law of Siemens. Here the ratio w/W is the measure of the efficiency of the motor, and the equation shows that we may make this efficiency as nearly equal to unity as we please, by letting the motor run so fast that e is very nearly equal to E: which is the true law of efficiency of a perfect motor supplied with electric energy, under the condition of constant external electromotive force.

Now go back to equation (3), which isw=Ec-c2R.

In order to find what value of c will give us the maximum value for to (which is the work done by the motor in unit time) we must take the differential coefficient and equate it to zero.

dw/dc=E-2cR = o; whence we have c = 1/2 E/R

But by Ohm's law, E/R is the value of the current when the motor stands still. So we see that, to get maximum work per second out of our motor, the motor must run at such a speed as to bring down the current to half the value which it would have if the motor were at rest. In fact, we here prove the law of Jacobi for the maximum rate of doing work. But here, sincec = E-e/R=1/2 E/R it follows that-

E-e = 1/2E; or c/E=1/2, whence it follows also thatw/W=1/2.

That is to say, the efficiency is but 50 per cent. when the motor does its work at the maximum rate.

Throughout it has been supposed that the motor is to be worked with a supply of current furnished at a fixed electromotive force. It is convenient and wise to make such a condition the basis of the argument, because this is probably the condition under which electric power will be distributed over large areas. It is true that this is not the only condition of supply, for a generator or system of generators may be worked so as to yield a constant current. And it would be quite possible to formulate a set of rules for the efficiency and maximum duty of motors under this condition. But this method of distributing electric power is far less likely to be of importance in the near future, than distribution with constant electromotive force; though for transmission of power to an isolated station, the case becomes of importance. One simple problem is worthy of mention. Suppose that one is desirous of working a motor so as to do work at the rate of a specified number of horse - power, and that the wire available to bring the current cannot safely stand more than a certain current without being in danger of becoming heated unduly; it might be desirable to know what electromotive force such a motor ought to be capable of giving back, and what electromotive force must be applied at the transmitting end of the wire.

Let N stand for the number of horse - power to be transmitted, and c for the maximum strength of current that the wire will stand (expressed in amperes). Then, by the known rule for the work of a current, since ec/746 = N e = 746 N / c gives the condition as to what electromotive force (in volts) the machine must be capable of giving, when run at the speed it is eventually to run at as a motor. Moreover, the primary electromotive force £ must be such that E - 6 / ∑R = c where ∑R is the sum of all the resistances in the circuit. Whence,

E = e + c∑R. Which is the required condition.

Another problem in the application of motors to transmission of power, which vitally affects their construction, is the determination of the relation of the heat - waste to the electromotive force at which the current is supplied to the motor.

If, as before, 2R stands for the sum of all the resistances in the circuit, then by Joule's law the heat - waste is (in mechanical measure)

HJ = C2∑R.

And since c = E-e / ∑R we may write the heat - waste as

HJ = (E-e)2 /∑R

Suppose that, without changing the resistances of the circuit, we can increase E, and also increase e, while keeping E - e the same as before, it is clear that the heat - loss will be precisely the same as before. But how about the work done? Let the two new values be respectively E' and e'. Then the electric energy expended is - W' = E' (E-e)