4* PRESSURE AND ENERGY 359 



as the number of particles with the velocities u, v, w which, in a 

 unit of time, cross the surface-element dy dz in the interior of the 

 space filled with gas. The upper sign applies to the passage in 

 the positive direction of x, and the lower to that in the opposite 

 direction. 



The analogous expressions for the other axial directions are 



+ NF(u, v, w)v du dv dw dz dx, 

 NF(u, v, w)w du dv dw dx dy. 



These formulae have a very simple interpretation. For u dy dz 

 is the volume of an oblique parallelepiped on the base dy dz, of 

 length equal to the velocity, and of altitude equal to the 

 component u ; it is thus the volume of the region in which all 

 the particles which cross dy dz in a unit of time with the given 

 velocity must have been at the beginning of the time-unit. Since, 

 from the definition of F, the first of the three expressions denotes 

 the number of particles in this volume which at any moment are 

 moving with the given velocity, it shows, as do also the other tw r o, 

 that as many particles pass through the surface-element as if none 

 had been previously stopped or deviated. 



The motion in a gas which is in the same state of equilibrium 

 at all points of the space occupied by it, therefore, goes on just as 

 if the particles never collided, but moved about in all directions 

 without hindrance. 



The reason for this (at first sight) surprising result is simply 

 that, when the requisite state of unchangeable equilibrium is 

 attained, for every particle which loses its motion by collision 

 there occurs another which acquires the same motion by another 

 simultaneous collision. 



5*. Momentum Carried Over 



The possibility thus demonstrated of replacing the hypotheses 

 on which our theory is founded by still simpler assumptions in 

 the case of a gas in equilibrium very considerably facilitates the 

 calculation of the pressure exerted by the gas. 



On the theory here assumed, the pressure exerted on a 



surface within the gaseous medium is measured by the force 



which one half of the medium exerts on the other from which 



it is separated by the surface. 1 Since, as has just been proved, 



1 See Chap. II. 12. 



