an increase of efficiency, whereas in reality the system has become less efficient. The variables in the above equations are v, v1, P, and P; the dimensions of the machines (or sliders) d and d1, and the intensities of the fields being constant. Since the ratio between the static efforts, P and P1, is also a constant, the number of variables is reduced to three, and, if two of these are given, the third can be found. As an example, we will take the case that the load P to be put on the receiving machine shall be given (say, for instance, the pull required to haul up a train on a steep gradient, but neglecting for the moment the difference in pull caused by variations of speed) and the speed v, of the generating machine shall also be given. We require to know the power necessary to drive the generating machine, and the speed and energy developed by the receiving machine. From the equation for P, we find immediately the speed of the receiving machine,

1

F1 d v = vi F d

r P F2 d2.

As will be seen, this speed is not directly proportional to the speed of the generator, and if the latter be increased the speed of the receiver will increase in a somewhat faster ratio. Since the ratio of speeds enters into the formula for the efficiency, it will be evidently advantageous to work the machines at the highest possible speed consistent with mechanical safety. On the other hand, if we lower the speed of the generator beyond a certain point the receiver will not be set in motion at all..

LOST ENERGY.

In this case the efficiency is zero.

45

The minimum speed

of the generator is therefore dependent on the dimensions of the two machines and on the strength of the two fields, and is inversely proportional to the product of these four quantities.

The mechanical energy which has to be applied to the

the difference between the two being lost.

which is represented by the expression r

This loss,

Р 2

(Pa);

we must regard as energy transformed in a way not suitable for the purpose in view. Since it does not appear in the shape of mechanical energy we must expect to find it appearing in the shape of heat, and this is indeed the case, as can easily be proved. It has been pointed out above that the static pull is the product of current, fieldintensity, and the dimension of the machine. The P

quotient represents, therefore, nothing else but the

F d

current flowing through the circuit, and the above term for the energy lost can also be written in the form

r c2,

which, as is well known, represents the heat developed by the passage of the current e through a circuit, the electrical resistance of which is r. Thus the whole of the energy applied at the generator is accounted for, partly by that given out by the receiver, and partly by that used up in heating the circuit. It need hardly be mentioned that the formulas given here for the transmission

of energy refer to ideal machines which are free from all other losses, both mechanical and electrical, but that in actual practice these other losses cannot be neglected, and considerably complicate the problems to be solved. The author, nevertheless, has thought it advisable to enter at some detail into the case of transmission of energy by means of ideal machines, not because the formulas obtained are immediately applicable to practical cases, but because they form the basis of other formulas suitably altered for practical purposes, and which will be given in a subsequent chapter. The example cited is also intended to show how easily and simply the system of absolute electro-magnetic measurement can be applied to apparently complicated problems. Before leaving this subject we must refer to the relation between electrical units in absolute measure and those units commonly used in practice. The units as given in the centimetergram-second system are of inconvenient magnitude for practical purposes; some of them are so small that millions and even larger figures are required to express quantities commonly dealt with in practical work, and others are, again, so large as to necessitate the use of fractions. We had already occasion to refer to the three units most often occurring in electro-mechanical problems, viz., current, electro-motive force, and resistance. The unit of quantity of electricity has also incidentally been mentioned as represented by that amount of electrical matter which a given current conveys in one second. For the sake of completing the list we must also mention a property of conductors called their capacity, by which term we mean their capacity or power to hold an electrical charge. The capacity is measured by the quantity of electricity with which a body can be charged under an

PRACTICAL UNITS.

47

electro-motive force equal to unity. The relation between the so-called practical units and their equivalents in the centimeter-gram-second system is as follows:—

CHAPTER II.

First Electro-motor-Professor Forbes' Dynamo-Ideal Alternating Cur rent Dynamo-Ideal Continuous Current Dynamo-Siemens' ShuttleWound Armature-Effect of Self-Induction-Experiments with Electromotors-Hefner-Alteneck Armature-Gramme Armature-Pacinotti Armature-Electro-motive Force created in any armature.

IN the preceding chapter it was shown how mechanical energy can be converted into that of an electric current, and how the electric energy represented by a current flowing under a given difference of potential can be reconverted again into mechanical energy and do useful work. The apparatus employed for this double conversion was assumed to be of extremely simple form, in order to limit our investigation to the fundamental laws without obscuring these laws by the introduction of secondary actions and losses. It will now be necessary to confront the subject from a somewhat more practical standpoint, and to show how the conversion between mechanical and electrical energy can be obtained with machinery of a practical form. As a first step towards a practical solution of the problem to produce motive power by an electric current, we must consider Barlow's wheel,' invented by Sturgeon and Barlow about sixty-five years ago. A star-shaped wheel was mounted on a horizontal

1 Barlow, "On Magnetic Attraction." London, 1823.