with Remarks upon the Mechanical Theory of Heat. 85 



more simply assume that all molecules move at the same rate ; 

 and for this case we obtain the following result : — 77^;? mean 

 lengths of path for the two cases (1) icliere the remaining molecules 

 move loith the same velocity as the one ivatclied, and (2) lohere 

 they are at rest, bear the proportion to one another of ^ to 1. 



It would not be difficult to prove the correctness of this rela- 

 tion : it is, however, unnecessary for us to devote our time to it ; 

 for, in our consideration of the mean path, it is not the question 

 to determine exactly its numerical value, but merely to obtain 

 an approximate notion of its magnitude ; and hence the exact 

 knowledge of this relation is not necessary. It is even suffi- 

 cient for our purpose if vre may assume as certain that the mean 

 path among moving molecules cannot be greater than among 

 stationary ones ; this will certainly be at once admitted. Under 

 this hypothesis, we will confine the discussion of the question 

 to that case where the molecule watched alone moves, while all 

 the others remain at rest. 



Moreover, without affecting the question in anything, we may 

 suppose a mere moving point in place of the moving molecule ; 

 for it is in fact only the centre of gravity of the molecule 

 which has to be considered. 



(5.) Suppose, then, there is a space containing a great num- 

 ber of molecules, and that these are not regularly arranged, the 

 only condition being that the density is the same throughout, i.e. 

 in equal parts of the space there are the same numbers of mole- 

 cules. The determination of the density may be performed con- 

 veniently for our investigation by knowing how far apart two 

 neighbouring molecules would be separated from one another 

 if the molecules were arranged cubically, that is, so arranged 

 that the whole space might be supposed divided into a number 

 of equal very small -cubic spaces, in whose corners the centres 

 of the molecules were situated. We shall denote this distance, 

 that is, the side of one of these little cubes, by \, and shall call 

 it the mean distance of the neighbouring molecules. 



If, now, a point moves through this space in a straight line, 

 let us suppose the space to be divided into parallel layers per- 

 pendicular to the motion of the point, and let us determine hoxo 

 great is the probability that the point will pass freely through a 

 layer of the thickness x without encountering the sphere of action 

 of a molecule. 



Let us first take a layer of the thickness 1, and let us denote 

 by the fraction of unity a the probability of the point passing 

 through this layer without meeting with any sphere of action : 

 then the corresponding probability for a thickness 2 is a"^ ; for if 

 such a layer be supposed divided into two layers of the thickness 

 1, tlic ])robability of the points passing free through the first 

 layer, and thereby arriving at the second, must be umltiplicd by 



