106 Prof. It. Clausius on the Theorem of the Mean Ergal, 



that it has for its numerator the entire volume V, while the new 

 one has only the part of the volume which is free for the motion 

 as its numerator. The latter is more precise ; but the difference 

 is so* slight that it may usually be neglected, because a character- 

 istic peculiarity of gases is that only a very small part of the 

 entire space occupied by a gas is actually filled by the spheres of 

 action— especially if the gas is a perfect gas ; for this idea itself 

 implies that, compared with the entire space, the space filled by 

 the spheres of action is a negligible quantity. After the fore- 

 going, however, there is no difficulty in taking this slight differ- 

 rence in the formulas into account; and in the above-men- 

 tioned proposition previously advanced we need only, instead of 

 the words " the entire space occupied by the gas/' to put u the 

 part of the space which is left free from the spheres of action of 

 the molecules." 



If the space furnished with stationary molecules has only the 

 magnitude V and is circumscribed by a firm envelope which the 

 point strikes against and recoils from, and if we wish to take 

 into consideration these collisions also, only a slight modification 

 of the formulae is necessary for this purpose. Let s be the mag- 

 nitude of the outer circumscribing surface; we have then for S 

 to put instead of N4<7r/) 2 the sum N47r/> 2 + 5, by which we get 



(N4y + s> 

 4(V-Nf7^p a ) , {/ } 



V — N^TTfl 3 



/f — 4 1 tl s^P ( 7Q\ 



'- 4 N47rp 2 + * {/6) 



With gases not very much rarefied, however, the surface s is 

 so small in comparison with N47T/9 2 that this difference also can 

 in most cases be neglected. 



In the foregoing we have assumed only one molecule as 

 moving and all the others as stationary. But in reality all the 

 molecules are in motion ; and it is inquired how in this case the 

 number of collisions suffered by a molecule during the unit of 

 time, and its mean length of path, can be determined. 



If we again imagine first the space which is filled with moving 

 molecules to be infinitely extended, so that the molecule consi- 

 dered strikes against other molecules only, and not against a 

 firm envelope, in order to determine the number of collisions 

 which it undergoes, we need only put in equation (70), in the 

 place of the absolute velocity of the molecule, its mean relative 

 velocity to all the other molecules ; this we will denote by r. We 

 thereby get for the number of collisions (which in this case may 

 be called P) the equation 



NTrpV 



