in the Kinetic Theory of Gases. 101 



at the time t + Bt assumes the value <£*, if for the variables the 

 values are substituted which correspond to the middle points 

 of do* and dm* ; then 



which quantity we have already denoted by 8'<f> in equation (8). 

 According to equation (9), 



B^cf) + 8 2 t(j> + S 3 2<£ = §B'<j>fdo dco. 

 If, now, we put <f> = lf, we have 



S^cfy + 8£<f> + B£cf>= §8'fdo dm = J/* do* d<o* - §fdo dm. 



Since in the last expression both the first and second inte- 

 grals express the total number of the molecules of the gas, 

 their difference must be zero. 



The indices and prefixed symbol B' have for the function/ 

 exactly the same meaning as before for the function <£. 



Hence, if we put (f> = lf, SS$ reduces to S 4 2<£ ; and we ob- 

 tain, if we make use of equations (4) and (18), the equation 



B^flfdo da> = ia 2 Bt j V cos 9 do dco dp dX . (ffr-ffj) . l (fff)' ( 22 > 



We will also make use of the general equation for the 

 change of the function /, which I have called equation (44) 

 in my " Further studies on the Heat-equilibrium among Gas- 

 molecules,"f and equation (2) in my paper " On the Heat- 

 equilibrium of Gases upon which External Forces act "J. 

 Although in the last-named paper I have proved this equation 

 at length, yet I will shortly deduce it from the data before us. 

 The number of molecules whose centres at the time t lie in 

 the parallelepiped do, and whose velocity-points at the same 

 time lie in the parallelepiped dco, is 



N l =f(%,y,z,Z,r),£,t)dodm, .... (23) 



for which, again, we will write for shortness fdo dm ; so also 

 the number of molecules for which, at the time t + Bt, the 

 centre lies in the parallelepiped do*, and the velocity-point in 

 the parallelepiped dm* : — 



N 2 =/(« + &t,y + nBt, z + $t, % + XBt, v + YBt, £+ ZBt, t + Bt)do* dm* 



=*w[/ + a<| +f |{, + ,| + ?|r + x| + Y^ +Z |)].(2 1 ) 



t Boltzmann, Wien. Sitxb. vol. lxvi. (1872). \ Ibid. vol. lxsii. (1875). 



