and its Application to the Molecular Motions of Gases. 105 



space contain equal numbers of molecules. Between these fixed 

 molecules one molecule only shall move, which, now here, now 

 there, shall strike against a fixed one and recoil from it. The 

 problem is, to find the number of collisions during the unit of 

 time; from which we shall then also obtain the mean length of 

 path. 



Instead of a moving molecule, we can consider a mere point in 

 motion, if we at the same time imagine the above-defined spheres 

 of action described about the centres of gravity of the fixed mo- 

 lecules, and assume that, as often as the moving point strikes 

 against the surface of a sphere of action, it recoils from it. We 

 thereby obtain again the case discussed in § 10, inasmuch as 

 the surface of each sphere of action forms a part of the surface 

 which limits the free space for the motion of the point. If the 

 space in which the fixed molecules are found is circumscribed 

 externally by a firm envelope, from which also the point recoils 

 when it strikes it, then the surfaces of the spheres of action and 

 the exterior bounding surface together constitute the entire sur- 

 face which limits the free space for the motion. 



We will first assume that the space furnished in the given manner 

 with fixed molecules is infinitely extended, so that the moving 

 point strikes against the surfaces of the spheres of action only, 

 and not against a firm envelope. The number of the stationary 

 molecules present may be determined by the statement that in 

 volume V their number is N. The spheres of action of these N 

 molecules occupy together the space NJ7T/3 3 ; so that the part of 

 the volume V remaining free for the motion of the point is repre- 

 sented by the difference V— Nj7iy) 3 . Further, the surfaces of 

 the spheres of action of the N molecules form together a surface 

 of the magnitude N47rp 2 . The two expressions V— Nf 7T/3 3 and 

 N47rp 2 we have to put in equations (68) and (69) in the places 

 of W and S ; we thus get 



N4tt P ^ _ Ntt^ 

 ^V-NfTr^'V-NlTrp 3 ' ' • ' ^ } 



V — l\ T4 -7rn 3 



'-^TrJ^- ; ™ 



In order to compare this expression of I 1 with that in (66), 



/-^ 



7rp z 

 we multiply numerator and denominator in the latter by N, and 

 then put (corresponding with the signification of X) NA, 3 = V, 

 which produces y 



/,= v 



This expression differs from that given in (71) by this only — 



