and its Application to the Molecular Motions of Gases. 197 



within a certain volume V by motions of material points in a 

 rectangularly parallelepipedal vessel, and employing coordinates 

 parallel to its sides and having their origin in its centre, we ex- 

 press the forces exerted on a point by the wall that is perpendi- 

 cular to the #-axis, not generally by F'(c— x) and —Y\c-\-x) f 

 but, as in § 5, by 



not 71 , noc n 



—-m-, ^zrr, and m- — -. 



(c—x) n+l {c + x) n+l 



We can then make use of Tall the formulae developed in §§ 6, 

 7, and 8, and can, indeed, further simplify them, by now as- 

 suming that the quantity a (which we there assume to be so 

 small that the force in the centre of the vessel was insensible) 

 is so small that the force is sensible only in the immediate 

 vicinity of the side — and therefore, in developing the series, 

 taking into account only the terms of the first order with re- 

 spect to u. 



Of the equations given in those sections we will select equa- 

 tions (53) and (54) for further discussion, but will abbreviate 

 them by omitting the terms which contain as a factor a 

 higher than the first power of /3, which is of the same order 

 as a. In this form they read : — 



. 2/3 £i! 



h= - - w » 

 n c 



log u= log (16wc 2 ) —2 -w~». 



We have previously advanced these equations for a single mo- 

 vable material point only; we will now derive from them equa- 

 tions valid for the entire system of points to be considered. 



For this purpose, in the former we substitute for w the pro- 

 duct tt^ 2 and multiply it by f(z)dz, thereby getting 



hf{z)dz=l£{wz*y^Mdz. 



Integrating this equation from z = to -s = x, we obtain on the 

 left-hand side the arithmetic mean of all the occurring values of 

 h, which, as before, we will denote by h- } this gives 



h = - - U> "^"l z* —f{z) dz. ... (103) 



We make the same substitution in the second of the above equa- 

 tions ; but we multiply it by z 2 f(z)dz, from which results : — 



logw . z*f(z)dz = log (16mc 2 ) . z i f{z)dz 



