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



cules, when of course, the individual values being various, 

 mean values must be taken. By way of example, I calculated 

 the case in which all the other molecules moved with exactly the 

 same velocity as the molecule under consideration, and deter- 

 mined the mean length of path (in this case designated by I) by 

 the equation \3 



l=i\ (67) 



The meaning of this equation, which I further transformed into 



I expressed in the following proposition*: — The mean length of 

 path of a molecule is to the radius of the spheres of action as the 

 entire space occupied by the gas is to that part of the space which 

 is actually filled by the spheres of action of the molecules. 



If we imagine the molecules as elastic balls, we must (as 

 already said) suppose the diameter of such a ball as great as the 

 radius of the above-defined sphere of action, whence we get for 

 the volume of the elastic ball one eighth of the volume of the 

 sphere of action. The last proposition then runs, The mean 

 length of path of a molecule is to one eighth of its diameter as 

 the entire space occupied by the gas is to that part of it which is 

 actually filled by the molecules. 



§ 10. To these previously found results on the collisions of 

 the molecules we will connect our further considerations. It 

 will, however, be advisable not to employ the formulae in their 

 previous shape, but to regard the subject in a somewhat different 

 manner, whereby we shall arrive at formulae which are still more 

 exact and have a form singularly suited to our present purpose. 



Given a space bounded by any irregular surface. At any 

 place whatever within this space let there be a moving point, so 

 that for all equally large portions of the space the probability of 

 their containing the point may be equal. Let the point make 

 an indefinitely small movement in any direction whatever, so 

 that all possible directions will be equally probable. Under 

 these circumstances, what is the probability that the point with its 

 indefinitely small movement, will strike the surface ? 



We will first consider a single element ds of the surface, and 

 inquire what is the probability that the point will strike precisely 

 this element of the surface. 



If dl is the indefinitely small extent of the point's motion, let 

 us now imagine the point at rest, and, conversely, that the sur- 

 face-element ds moves in the opposite direction about the portion 

 dl. Thereby an indefinitely small prismatic space will be de- 



* P°gg- <4*Mt. vol. cv. p. 260; Abhandlungensammlwng, vol. ii. p. 272 



