Introduction. [Equation 3]

velocity $c_2$ with different components $ξ_2, η_2, ζ_2$, and perhaps also a different mass $m_2$. An analogous meaning applies to the quantities $n_3, c_3, ξ_3, η_3, ζ_3, m_3$ and so on up to $n_{ν}, c_{ν}, ξ_{ν}, η_{ν}, ζ_{ν}, m_{ν}$. The state of the gas within the vessel shall remain stationaryIn thermodynamics, a stationary state means the macroscopic properties like pressure and temperature do not change over time, even though individual particles are moving. during the time $t$. Thus, even if during any time interval $τ$ some of the $n_1 Ω$ molecules lose their velocity components $ξ_1, η_1, ζ_1$ through collisions with other molecules or with the vessel wall, on average an equal number of identical molecules will regain those same velocity components through collisions during that same time.

We must now first calculate how many of our $n_1 Ω$ molecules strike the piston on average during the time interval $t$. All $n_1 Ω$ molecules travel a distance $c_1 dt$ during a very short time $dt$ in such a direction that its projections onto the coordinate axes are $ξ_1 dt, η_1 dt,$ and $ζ_1 dt$. If $ξ_1$ is negative, the molecules in question cannot strike the piston A negative value implies the molecule is moving away from the piston face.. If, however, it is positive, we construct a slanted cylinder within the vessel, the base of which is the piston $A B$, and the side of which is equal to and in the same direction as the path $c_1 dt$. Then, those and only those of our $n_1 Ω$ molecules that were located in this cylinder at the beginning of the moment $dt$—the number of which we shall designate as $dν$—will collide with the piston during the time $dt$. The $n_1 Ω$ molecules are, on average, uniformly distributed throughout the entire vessel, and this uniform distribution extends right up to the vessel walls, since the molecules reflected by them move back exactly as if the vessel walls were not there and there were an identical gas on the other side. Therefore, the ratio of $n_1 Ω$ to $dν$ is the same as the ratio of $Ω$ to the volume of the slanted cylinder ¹); the latter is equal to $\varphi ξ_1 dt,$ from which it follows:

Since the state in the vessel now remains stationary, during any time $t$, of our $n_1 Ω$ molecules

1) Regarding the conditions for the validity of an analogous proportion, cf. § 3.