Such is the value of t obtained on leaving all the units undetermined. If we express ρ as a function of the second, we have t in seconds. If we take ρ = 1, we have the absolute value Θ of the same interval of time as a function of this unit; we have simply Θ = pq.
If we suppose that the commutator which produces the successive charges and discharges of the condenser consists of a vibrating tuning fork, we see that the duration of a vibration is equal to the product of the two abstract numbers, pq.
It remains for us to ascertain to what degree of approximation we can determine p and q. To find q we must first construct a column of mercury of known dimensions; this problem was solved by the International Bureau of Weights and Measures for the construction of the legal ohm. The legal ohm is supposed to have a resistance equal to 106.00 times that of a cube of mercury of 0.01 meter, side measurement. The approximation obtained is comprised between 1/50000 and 1/200000. To obtain p, we must be able to construct a plane condenser of known capacity. The difficulty here consists in knowing with a sufficient approximation the thickness of the stratum of air. We may employ as armatures two surfaces of glass, ground optically, silvered to render them conductive, but so slightly as to obtain by transparence Fizeau's interference rings. Fizeau's method will then permit us to arrive at a close approximation. In fine, then, we may, a priori, hope to reach an approximation of one hundred-thousandth of the value of pq.
Independently of the use which may be made of it for measuring time in absolute value, the apparatus described possesses peculiar properties. It constitutes a kind of clock which indicates, registers, and, if needful, corrects automatically its own variations of speed. The apparatus being regulated so that the magnetic needle may be at zero, if the speed of the commutator is slightly increased, the equilibrium is disturbed and the magnetic needle deviates in the corresponding direction; if on the contrary the speed diminishes, the action of the antagonistic circuit predominates, and the needle deviates in the contrary direction. These deviations, when small, are proportional to the variations of speed. They may be, in the first place, observed. They may, further, be registered, either photographically or by employing a Redier apparatus, like that which M. Mascart has adapted to his quadrant electrometer; finally, we may arrange the Redier to react upon the speed so as to reduce its variations to zero. If these variations are not completely annulled, they will still be registered and can be taken into account.
As an indicator of variations this apparatus can be of remarkable sensitiveness, which may be increased indefinitely by enlarging its dimensions.
With a battery of 10 volts, a condenser of a microfarad, 10 discharges per second, and a Thomson's differential galvanometer sensitive to 10-10 amperes, we obtain already a sensitiveness of 1/1000000, i.e., a variation of 1/1000000 in the speed is shown after some seconds of a deviation of one millimeter. Even the stroboscopic method does not admit of such sensitiveness.
We may therefore find, with a very close approximation, a speed always the same on condition that the solid parts of the apparatus (the condenser and the resistance) are protected from causes of variation and used always at the same temperature. Doubtless, a well-constructed astronomical clock maintains a very uniform movement; but the electric clock is placed in better conditions for invariability, for all the parts are massive and immovable; they are merely required to remain unchanged, and there is no question of the wear and tear of wheel-work, the oxidation of oils, or the variations of weight. In other words, the system formed by a condenser and a resistance constitutes a standard of time easy of preservation.
In this system the measurement of time is not effected, as ordinarily, by observing the movements of a material system, but by experiments of equilibrium. All the parts of the apparatus remain immovable, the electricity alone being in motion. Such appliances are in a manner clepsydræ. This analogy with the clepsydræ will be perceived if we consider the form of the following experiment: Two immovable metallic plates constitute the armatures of a charged condenser, and attract each other with a force, F. If the plates are insulated, these charges remain constant, as well as the force, F. If, on the contrary, we connect the armatures of resistance, R, their charges diminish and the force, F, becomes a function of the time, t; the time, t, inversely becomes a function of P. We find t by the following formula:
t = ρ × (lS / Sπes) × log hyp(F0/F)