158 Prof. Ludwig Boltzmann on the 



components of the velocity resolved along the coordinate 

 axes. 



We may denote the position of the condition-point at time 

 t + St by A'. A / will be called the point corresponding to A, 

 If we consider St as constant, every point within the condition- 

 cylinder will possess its corresponding point. Always, when 

 at any time the condition-point has occupied any point of 

 space, it will, after a time St has elapsed, pass to the corre- 

 sponding point ; and vice versa, the moving body can never 

 reach the corresponding point except by having passed 

 through the point to which it corresponds at a time St 

 previously. 



We have then 



x' = x + gcos 6 . St, y'=y+.q sin 6 . St, I n , 



u' = u + %.St, v'=v + y.8t, I 



in which £= — ^— , tj= — ^— are the components of the force 



acting on the moving particle, and are therefore functions of 

 as and y. Further, 



0'=arctff-7> 

 ^u' 



which gives, on substituting the above values, 



e'^e+iv.cose-Zsme)™. ... (2) 



Now let us construct an infinitely small rectangular 

 parallel opiped dxdydO, of which one edge is situated at the 

 point A. Let the fraction of the whole time T during which 

 the condition-point lies within the parallelopiped be dt. This 

 is the time during which the three variables x, y, will be 

 included between the values x and x + dx, y and y + dy, and 

 6 and 6 + d6. We may then write in all cases 



dt = f(x,y,6)dxdydd (3) 



Now construct at each point of the parallelopiped dxdydO 

 the corresponding point, and hence obtain the parallelopiped 

 dx' dy' d&. That fraction of the time T during which the 

 condition-point lies within dx dy' dQ' is, according to equa- 

 tion (3), 



dt'=f{x',y', 0') dx' dy' dd> '. 



Since, however, according to our definition of the condition- 

 point, every time it enters the parallelopiped dxdydO it must 

 after an interval St enter dx' dy' d& , and since the interval 



