We have then

w
E1 = -------- f
[Omega]                            (1.)

If R be the resistance in circuit by Ohm’s law,

E — E1
C = --------
R

w
= E  ------- f(C)
[Omega]
----------------
R

and therefore

[Omega](E — CR)                        (2.)
w = -----------------
f(C)

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

Power required = L w = a E1 C

w
= a C ------- f(C)
[Omega]

Dividing by w,

a C f(C)
L = -------- .                            (3.)
[Omega]

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