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, [rho], of mercury in absolute electrostatic
units; [rho] is a specific property of mercury, and is consequently a
magnitude absolutely invariable. Moreover, [rho] is _an interval of
time_. We might, therefore, take [rho] as a unit of time, unless we
prefer to consider this value as an imperishable standard of time.
In fact, [rho] 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, [rho] 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 [rho]. 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 [rho] 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 [rho]. This problem may be solved in various ways,[1] and
especially by means of the following apparatus.
[Footnote 1: 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 = [rho] × (lS / S[pi]es) × log hyp(F0/F)
F0 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 [rho] the corresponding value ([rho] = 1.058 X
10^-16). If we take [rho] as a unit we make [rho] = 1, and we
find the absolute value of the time by the expression:
(lS) / (8[pi]es) log hyp(F0/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.]
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,
[tau].
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, [tau], the continuous current yields a
E
quantity of electricity = -- [tau]; on the other hand, each charge of
R
the condenser = CE, and during the time, [tau], the number of
[tau]
discharges = -----, t being the fixed time between two discharges;
t
[tau] and t are here supposed to be expressed by the aid of an
arbitrary unit of time; the second circuit yields, therefore, a
[tau]
quantity of electricity equal to CE × -----. The condition of
t
E [tau]
equilibrium is then ---[tau] = CE × ----- ; or, more simply, t = CR.
R t
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
l [rho]
have, therefore, R = q[rho] --- = q ----- ; and consequently t = pq[rho].
l² l
Such is the value of _t_ obtained on leaving all the units
undetermined. If we express [rho] as a function of the second, we have
_t_ in seconds. If we take [rho] = 1, we have the absolute value
[Theta] of the same interval of time as a function of this unit; we
have simply [Theta] = _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.
* * * * *