We have then

E1 = w/Ω f . (1)

If R be the resistance in circuit by Ohm's law.

C= E-E / R

= E-w/Ω f(C); R and therefore w= Ω(E - CR)/f(C) . (2)

Let a be the effciency with which the motor transforms electrical into mechanical energy, then -

Power required = Lw = aE1 C

= aC w/Ω f (c)

Dividing by w,

L = aCf (c) / Ω (3)

It must be noted that L is here measured in electrical measure, or, adopting the unit given by Dr. Siemens, 1 Joule equals approximately 0.74 foot - lb. 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 OR 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 the circuit; and secondly, one or more low resistances for varying the speed of the car. If the form of /(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 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 the dynamo revolving at a high speed, the momentum of the current is 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 15 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 break-ing down the insulation of one of the wires of the magnet coils. The way out of the difficulty is 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 Port - rush 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 con stant 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 2 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 4 cars are running, requiring 4 h. - p. each, the loss due to resistance does not exceed 4 Per cent. of the power developed on the cars; or if 1 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.

The insulation of the conductor has been a matter of considerable difficulty, chiefly on account of the moistness of the climate. An insulation has now, however, been obtained of 500 to 1000 ohms per mile, according to the state of the weather, by placing a cap of insulite between the wooden posts and T - iron. Hence the total leakage cannot exceed 2.5 amperes, representing a loss of f 2/4. - p., or under 5 per cent., when 4 cars are running.

Apart from these figures, we have materials fur an actual comparison of the cost of working the line by electricity and steam. The steam tramway engines, temporarily employed at Portrush, are generally considered as satisfactory as any of the various tramway engines. They have a pair of vertical cylinders, 8 in. diameter and 1 ft. stroke, and work at a boiler pressure of 120 lb., the total weight of the engine being 7 tons. The electrical car with which the comparison is made, has a dynamo weighing 13 cwt., and the tare of the car is 52 cwt. The steam - engines are capable of drawing a total load of about 12 tons up the hill, excluding the weight of the engine; the dynamo over 6 tons, excluding its own weight; hence, weight for weight, the dynamo will draw 5 times as much as the steam - engine. Finally, compare the following estimates of cost. From actual experience, the steam - engine, taking an average over a week, costs -

The distance run was 312 miles. Also, from actual experience, the electrical car, drawing a second behind it, and hence providing for the same number of passengers, consumed 18 lb. of coke per mile run. Hence, calculating the cost in the same way, for a distance run of 312 miles in a week -

A saving of over 25 per cent.

The total mileage run is very small, on account of the light traffic early in the year. Heavier traffic will tell very much in favour of the electric car, as the loss due to leakage will be a much smaller proportion of the total power developed.

It will be observed that the cost of the tramway engines is very much in excess of what is usual on other lines, but this is entirely accounted for by the high price of coke, and the exceedingly difficult nature of the line to work, on account of the curves and gradients. These causes send up the cost of electrical working in the same ratio, hence the comparison is valid as between the steam and electricity, but it would be unsafe to compare the cost of either with horse traction or wire - rope traction on other lines. The same fuel was burnt in the stationary steam - engine and in the tramway engines, and the same rolling stock used in both cases; but otherwise the comparison was made tinder circumstances in favour of the tramway engine, as the stationary steam - engine is by no means economical, consuming at least 5 lb. of coke per horse - power hour, and the experiments were made, in the case of the electrical car, over a portion of line 3 miles long, which included the worst hills and curves, and one - half of the conductor was not provided with the insulite caps, the leakage consequently being considerably larger than it will be eventually.

Finally, as regards the speed of the electrical car, it is capable of running on the level at the rate of 12 miles per hour.

Taking these data as to cost, and remembering how this will be reduced when the water - power is made available, added to such considerations as the freedom from smoke and steam, the diminished wear and tear of the permanent way, and the advantage of having each car independent, it may be said that there is a future for electrical railways. (Dr. Hopkinson.)