A force "F" will be exerted on the piston, equal to  According to Newton's second law of motion, this force will create an acceleration "a" equal to As this process is adiabatic, the equation for ideal gas (Poisson's equation) is: It follows using differentiation from the equation above that:  If the piston moves by a distance formula_6 in the glass tube, the corresponding change in volume will be By substituting equation into equation , we can rewrite as follows: Solving this equation and rearranging terms yields the differential equation of a harmonic oscillation from which the angular frequency ω can be deduced:  From this, the period "T" of harmonic oscillation performed by the ball is: Measuring the period of oscillation "T" and the relative pressure "P" in the tube yields the equation for the adiabatic exponent:  In 1929, Rinkel proposed a different method to calculate formula_3 while using the Rüchardt apparatus: he noted that it may be shown that the vertical distance "L" which the sphere falls before it begin to rise is: formula_8, so formula_3 may be calculated from measured values of "L", "m", "V", "P" and "A". In 1951, Koehler and later, in 1972 Flammersfeld introduced a trick in the original Rüchardt setup, to increase the number of oscillations that are limited by the unavoidable friction-damping and gas leak (through the piston-tube seal): they made a thin hole on the tube (at half-height) and provided a gas-feeding pump to keep the pressure inside the vessel constant. By properly trimming the gas inlet flux (through a throttling valve) they obtained the following result: during the oscillations the piston is pushed-up by the gas overpressure until it crosses the hole position; then the gas leakage through the hole reduces the pressure, and the piston falls back. The force acting onto the piston varies at a rate that is regulated by the piston oscillation frequency leading to forced oscillation; fine adjustment of the throttle valve allows to achieve maximum amplitude at resonance. In 1958, Christy and Rieser used only a gas-feeding pump to stabilize the gas pressure. A slightly different solution was found in 1964 by Hafner who used a tapered tube (conical: slightly larger at the top). In 1959, Taylor used a column of mercury oscillating inside a U-shaped tube instead of the Rüchardt sphere.