Let a be the efficiency with which the motor transforms electrical into mechanical energy, then—

Power required = L w =a E1C
w
=a C—f(C)
Ω

Dividing by w,

aCf(C)
L =————. (3.)
Ω

It must be noted that L is here measured in electrical measure, or, adopting the unit given by Dr. Siemens in the British Association Address, in joules. One joule equals approximately 0.74 foot pound. Equation 3 gives at once an analytical proof of the second principle stated above, that for a given motor the current depends upon the couple, and upon it alone. Equation 2 shows that with a given load the speed depends upon E, the electromotive force of the main, and R the resistance in circuit. It shows also the effect of putting into the circuit the resistance frames placed beneath the car. If R be increased, until CR is equal to E, then w vanishes, and the car remains at rest. If R be still further increased, Ohm's law applies, and the current diminishes. Hence suitable resistances are, first, a high resistance for diminishing the current, and consequently, the sparking at making and breaking of of the circuit; and, secondly, one or more low resistances for varying the speed of the car. If the form of f(C) be known, as is the case with a Siemens machine, equations 2 and 3 can be completely solved for w and C, giving the current and speed in terms of L, E, and R. The expressions so obtained are not without interest, and agree with the results of experiment.

It may be observed that an arc light presents the converse case to a motor. The E.M.F. of the arc is approximately constant, whatever the intensity of the current passing between the carbons; and the current depends entirely on the resistance in circuit. Hence the instability of an arc produced by machines of low internal resistance, unless compensated by considerable resistance in the leads.

The following experiment shows in a striking form the principles just considered: An Edison lamp is placed in parallel circuit with a small dynamo machine, used as a motor. The Prony brake on the pulley of the dynamo is quite slack, allowing it to revolve freely. Now let the lamp and dynamo be coupled to the generator running at full speed. First, the lamp glows, in a moment it again becomes dark, then, as the dynamo gets up speed, glows again. If the brake be screwed up tight, the lamp once more becomes dark. The explanation is simple. Owing to the coefficient of self-induction of the dynamo machine being considerable, it takes a finite time for the current to obtain an appreciable intensity, but the lamp having no self-induction, the current at once passes through it, and causes it to glow. Secondly, the electrical inertia of the dynamo being overcome, it must draw a large current to produce the kinetic energy of rotation, i.e., to overcome its mechanical inertia; the lamp is therefore practically short-circuited, and ceases to glow. When once the rotation has been established, the current through the dynamo becomes very small, having no work to do except to overcome the friction of the bearings, hence the lamp again glows. Finally, by screwing up the brake, the current through the dynamo is increased, and the lamp again short-circuited.

It has often been pointed out that reversal of the motor on the car would be a most effective brake. This is certainly true; but, at the same time, it is a brake that should not be used except in cases of emergency. For this reason, the dynamo revolving at a high speed, the momentum of the current is very considerable; hence, owing to the self-induction of the machine, a sudden reversal will tend to break down the insulation at any weak point of the machine. The action is analogous to the spark produced by a Ruhmkorff coil. This was illustrated at Portrush; when the car was running perhaps fifteen miles an hour, the current was suddenly reversed. The car came to a standstill in little more than its own length, but at the expense of breaking down the insulation of one of the wires of the magnet coils. The way out of the difficulty is evidently at the moment of reversal to insert a high resistance to diminish the momentum of the current.

In determining the proper dimensions of a conductor for railway purposes, Sir William Thomson's law should properly apply. But on a line where the gradients and traffic are very irregular, it is difficult to estimate the average current, and the desirability of having the rail mechanically strong, and of such low resistance that the potential shall not vary very materially throughout its length, becomes more important than the economic considerations involved in Sir William Thomson's law. At Portrush the resistance of a mile, including the return by earth and the ground rails, is actually about 0.23 ohm. If calculated from the section of the iron, it would be 0.15 ohm, the difference being accounted for by the resistance of the copper loops, and occasional imperfect contacts. The E.M.F. at which the conductor is maintained is about 225 volts, which is well within the limit of perfect safety assigned by Sir William Thomson and Dr. Siemens. At the same time the shock received by touching the iron is sufficient to be unpleasant, and hence is some protection against the conductor being tampered with.

Consider a car requiring a given constant current; evidently the maximum loss due to resistance will occur when the car is at the middle point of the line, and will then be one-fourth of the total resistance of the line, provided the two extremities are maintained by the generators at the same potential. Again, by integration, the mean resistance can be shown to be one-sixth of the resistance of the line. Applying these figures, and assuming four cars are running, requiring 4 horse power each, the loss due to resistance does not exceed 4 per cent. of the power developed on the cars; or if one car only be running, the loss is less than 1 per cent. But in actual practice at Portrush even these estimates are too high, as the generators are placed at the bottom of the hills, and the middle portion of the line is more or less level, hence the minimum current is required when the resistance is at its maximum value.