104 Prof. R. Clausius on the Theorem of the Mean Ergal, 



The same probability which holds for one surface-element, 

 holds also for every other of equal magnitude. We get there- 

 fore at once for any finite portion s of the surface selected for 

 consideration, or also for the entire surface (which shall be de- 

 noted by S), the following proposition : — -If within a space W 

 bounded by the surface S a point, starting from any position what- 

 ever, traverses in any direction the infinitesimal distance dl, the 

 probability that in its motion it will strike a certain part s of the 

 surface is represented by I 



m dl > 



and the probability that it will strike the surface generally is repre- 

 sented by g 



m dL 



We will now assume that the point not merely traverses the 

 infinitesimal distance dl, but, with a certain velocity v, continues 

 to move until it strikes the surface and, according to the laws of 

 elasticity, recoils, upon which its motion is continued with the 

 same velocity in another direction. We will, besides, presup- 

 pose that the force exerted on the point by the surface acts only 

 in its immediate vicinity, so that the alteration of the direction 

 of motion at the collision takes place in an imperceptibly short 

 time, and accordingly the velocity may be regarded as constant, 

 notwithstanding the variation during the collision. 



We can then, in the preceding proposition, replace the ele- 

 ment of path dl by the product vdt, and say, The probability 

 that during the infinitesimal time dt the point will strike the sur- 

 face is represented by 



Sv u 

 4W 



Thence results, for the average number of collisions during 

 unit time, which we will denote by P', the equation 



p'=w j ^ 



and for the mean length of path, /', we get, dividing v by P', the 

 equation 



4W 



'«nr (69 > 



§ 11. Let us now turn to the consideration of the molecules. 

 Instead, however, of assuming all the molecules alike to be in 

 motion, let us imagine a space in which very many molecules are 

 in fixed positions — not, indeed, regularly arranged, but yet so 

 far uniformly distributed that equal measurable sections of the 



