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 x -----.  The condition of
t
E [tau]
equilibrium is then ---[tau] = CE x ----- ; 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].
l squared 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.