2* PRESSURE AND ENERGY 357 



Take the axis of x perpendicular to the plane, and investigate 

 the number of those particles which in a unit of time cross an 

 infinitely small rectangle dy dz at the point (x, y, z) with a velocity 

 the components of which are u, v, w. 



Since this specification of the components determines the 

 direction of the motion of the particles in question, it follows that 

 all these particles must come from a limited region, an oblique 

 parallelepiped in shape, which has the rectangle dy dz as base and 

 its length along the direction of motion. 



Denote the coordinates of any point in this region relatively 

 to the given point (x, y, z) by r, p, j ; these must satisfy the 

 condition 



F : 9 : 5 = u : v : w, 



and the absolute coordinates of the point are x r, y \), z j. 

 Divide up also this region, from which all the molecules in 

 question come, into infinitely small oblique parallelepipeds 



cZ]C dy d% = d\ dy dz 



by planes drawn parallel to that of yz. 



In one of these volume-elements it will, by 1*, happen 



NF(u t v, w)f(t, u t v, w) -du dv dw dt dy dz 

 t 



times per time-unit that a particle begins a new path, with a 

 velocity whose components are u, v, w, in the direction towards 

 the surface-element dy dz, this path ending after the lapse of 

 time t. 



The particles will actually reach their mark, the element dy dz, 

 and pass through it if the time t is sufficient for the path to be 

 traversed, and therefore, in the case of particles moving in the 

 positive direction of x, if 



X > ut. 



We consequently obtain the total number of the particles which 

 start from the elements of that oblique parallelepiped standing on 

 the base dy dz, and pass through the element dy dz in unit time 

 with a velocity whose components are u, v, w, by summing up the 

 above expression for all the volume-elements with the condition 



X }> ut, 



that is, by integrating it with respect to dp from the initial value 

 y = to the limiting value = ut. In the second place, to obtain 



