By M. LIPPMANN.
The unit of time universally adopted, the second, undergoes only very slow secular variations, and can be determined with a precision and an ease which compel its employment. Still it is true that the second is an arbitrary and a variable unit - arbitrary, in as far as it has no relation with the properties of matter, with physical constants; variable, since the duration of the diurnal movement undergoes causes of secular perturbation, some of which, such as the friction of the tides, are not as yet calculable.
We may ask if it is possible to define an absolutely invariable unit of time; it would be desirable to determine with sufficient precision, if only once in a century, the relation of the second to such a unit, so that we might verify the variations of the second indirectly and independently of any astronomical hypothesis.
Now, the study of certain electrical phenomena furnishes a unit of time which is absolutely invariable, as this magnitude is a specific constant. Let us consider a conductive substance which may always be found identical with itself, and to fix our ideas let us choose mercury, taken at the temperature of 0° C., which completely fulfills this condition. We may determine by several methods the specific electric resistance, ρ, of mercury in absolute electrostatic units; ρ is a specific property of mercury, and is consequently a magnitude absolutely invariable. Moreover, ρ is an interval of time. We might, therefore, take ρ as a unit of time, unless we prefer to consider this value as an imperishable standard of time.
In fact, ρ is not simply a quantity the measure of which is found to be in relation with the measure of time. It is a concrete interval of time, disregarding every convention established with reference to measures and every selection of unit. It may at first sight, appear singular that an interval of time is found in a manner hidden under the designation electric resistance. But we need merely call to mind that in the electrostatic system the intensities of the current are speeds of efflux and that the resistances are times, i.e., the times necessary for the efflux of the electricity under given conditions. We must, in particular, remember what is meant by the specific resistance, ρ of mercury in the electrostatic system. If we consider a circuit having a resistance equal to that of a cube of mercury, the side of which = the unit of length, the circuit being submitted to an electromotive force equal to unity, this circuit will take a given time to be traversed by the unit quantity of electricity, and this time is precisely ρ. It must be remarked that the selection of the unit of length, like that of the unit of mass, is indifferent, for the different units brought here into play depend on it in such a manner that ρ is not affected.
It is now required to bring this definition experimentally into action, i.e., to realize an interval of time which may be a known multiple of ρ. This problem may be solved in various ways,1 and especially by means of the following apparatus.
A battery of an arbitrary electromotive force, E, actuates at the same time the two antagonistic circuits of a differential galvanometer. In the first circuit, which has a resistance, R, the battery sends a continuous current of the intensity, I; in the second circuit the battery sends a discontinuous series of discharges, obtained by charging periodically by means of the battery a condenser of the capacity, C, which is then discharged through this second circuit. The needle of the galvanometer remains in equilibrium if the two currents yield equal quantities of electricity during one and the same time, τ.
Let us suppose this condition of equilibrium realized and the needle remaining motionless at zero; it is easy to write the conditions of equilibrium. During the time, τ, the continuous current yields a quantity of electricity = (E / R)τ; on the other hand, each charge of the condenser = CE, and during the time, τ, the number of discharges = τ/t, t being the fixed time between two discharges; τ and t are here supposed to be expressed by the aid of an arbitrary unit of time; the second circuit yields, therefore, a quantity of electricity equal to CE × (τ / t). The condition of equilibrium is then (E / R) τ = CE × (τ / t); or, more simply, t = CR.
C and R are known in absolute values, i.e., we know that C is equal to p times the capacity of a sphere of the radius, l; we have, therefore, C = pl; in the same manner we know that R is equal to q times the resistance of a cube of mercury having l for its side. We have, therefore, R = q ρ (l / l²) = q (ρ / l) and consequently t = pqρ.
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 clepsydrae. This analogy with the clepsydrae 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(F/F)
F and F being the values of the force at the beginning and at the end of the time, t. The above formula is independent of the choice of units. If we wish t to be expressed in seconds, we must give ρ the corresponding value (ρ = 1.058 × 10-16). If we take ρ as a unit we make ρ = 1, and we find the absolute value of the time by the expression:
(lS) / (8πe s) log hyp(F/F)
We remark that this expression of time contains only abstract numbers, being independent of the choice of the units of length and force. S and e denote surface and the thickness of the condenser; s and l the section and the length of a column of mercury of the resistance, R. This form of apparatus enables us practically to measure the notable values of t only if the value of the resistance, R, is enormous, the arrangement described in the text has not the same inconvenience.