and its Application to the Molecular Motions of Gases. 35 

 from which comes 



Kt)'=*-(^r. < 2 " 



k being a constant the meaning of which is evident from this, 

 that mk represents the portion of the energy referred to the 

 ^-direction. This equation we can bring into the following 

 form : — 



A* dx (22) 



In order, further, to determine from this the time required for 



the point, moving from a place where # = 0, to arrive at the 



dx 

 place near the positive boundary-plane where 77 = an d where 



it reverses the direction of its motion, we have to integrate this 

 expression of the time-differential from x = to x=a, when a 

 denotes that value of x for which the equation 



k-j-^—^0 (23) 



(c — a) n v ' 



holds good. 



To effect the integration let us develop into a series the ex- 

 pression on the right side of equation (22), thus — 



y *T 1 + i «" - + hl ** + 1 



V2&L t 2i(c-^' + 2.4F(c-^ T J 



Integrating this expression from x=0 to x = a, and denoting 



the time thence obtained by -i, because it is one fourth of that 



which the point occupies in going and returning once between 

 the two sides, we get 



1;_ 1_ f" 1 u n 1 a" 



4 " Vn L tt+ 2 {n-\)k(c-a) n - 1 2 \n-\)kc n - x 



^3 ft »» _ 1^3 « 2 " _ \ 



+ 2 . 4 {2n-l)k\c-af n - l+ " ' 2.4 fin-l)**^- 1 "J 

 If in this we neglect the terms which have the minus sign, on 

 account of the smallness of the fraction -, and, in conformity 

 with (23), in the other terms put 



_i 1 



c—a = ak n and a = c — uk'~n 



D2 



