Introduction. [Equation 1]

be understood as a collision, and the first of these will exert the force $q_1$, the second the force $q_2$, and so on, upon the piston in the positive direction of the x-axis original: "Abscissenrichtung," referring to the horizontal coordinate in a Cartesian system. If we designate the mass of the piston as $M$ and its velocity in the positive x-direction as $U$, then for the time element $dt$ one has the equation:

If one multiplies by $dt$ and integrates over an arbitrary time $t$, it follows that:

Now, if $P$ is to be equal to the pressure of the gas, the piston must not fall into any noticeable motion, apart from invisible fluctuations. In the formula above, $U_0$ is the value of its velocity in the x-direction at the beginning of the time interval, and $U_1$ is the value of the same quantity after the lapse of time $t$. Both quantities will be very small; indeed, one can easily choose the time $t$ such that $U_1 = U_0$, since the piston, given its small fluctuations, must periodically assume the same velocity. In any case, $U_1 - U_0$ cannot increase as time increases; therefore, the quotient $(U_1 - U_0) / t$ must approach the limit of zero as time increases. From this it follows:

The pressure is therefore the average value original: "Mittelwerth" of the sum of all the small pressures which the individual colliding molecules exert on the piston at different times. We now wish to calculate $\int q dt$ for any single collision that the piston original: "Stempel," used here interchangeably with "Kolben" (piston) experiences from a molecule during the time $t$. Let the mass of the molecule be $m$, and let its velocity component in the positive x-direction be $u$. Let the collision begin at time $t_1$ and end at time $t_1 + τ$; the molecule then exerts no force at all upon the piston before time $t_1$ or after time $t_1 + τ$. It is therefore: