314 Prof. L. Boltzmann on the Assumptions necessary 



Integrating this expression for all the other differentials up 

 to dv and multiplying by 6, we have 



IT 



T$=S<jr 2 v 2 f{v,t)dvB 2 0\ \ \ i Y 2 F{Y,t)rrs<rdYdTdSdO.(9) 



Jo ^ o ^o Jo 



The kind and mode of impact is completely determined by 

 the values of the variables v, V, T, S, and 0. The magnitudes 

 v' and V' of the velocities of both molecules after impact, their 

 angle T' as well as the angle 0' which the plane ROB/ of the 

 two relative velocities makes with the plane v'Sl'Y' of the two 

 absolute velocities v' and V' after the impact, may therefore 

 be expressed as functions of v, Y, T, S, and 0. For all impacts 

 for which the latter variables lie between the limits (2), 

 v', Y f 9 T', S', and 0' will also lie between certain infinitely 

 close limits, which we may denote by 



v' and v' + dv f , Y' and V + dV', 



T' and T' + dT, 0' and O' + dO', S and S + dS. . (10) 



The angle S has the same meaning before and after impact, 

 and is therefore after impact also included within the limits S 

 and S + dS. But it is now clear that each impact may also 

 be taken in inverted order. If, therefore, inversely the values 

 of the variables before impact lay between the limits (10), 

 then they would also after impact lie between the limits (2) . 

 Exactly as with expression (3), so also will 



d7J^8ir 2 v ,2 Y' 2 f{y', i)F(V', ^er&T'so-dv'dY'dTdO'dW (11) 



be the number of impacts which, in the unit volume during 

 the time 0, occur between a molecule of the first order and 

 one of the second, so that after impact the variables shall lie 

 between the limits (2). Since v', Y', T', and CK are known 

 as functions of v, V, T, O, and S, we may here again intro- 

 duce the differentials of the latter variables, and obtain]) 



d7J = 8ir 2 v f2 Y /2 f(v r , t) F ( V ', t) 6r8 2 T'saA dv dY dTdOdS, (12) 

 where 



A _- , V BV BT' BO' 

 zx " z± o^"5V'BT'BO' • • • • ( i6 > 



the partial differentials being taken under the supposition 

 that 8 is constant. If we integrate expression (12) for all 

 variables except dv, we obtain all impacts which occur in unit 

 volume during the time 6 between a molecule of the first and 

 a molecule of the second kind, so that after impact the velo- 

 cities of the molecules of the first kind lie between the limits (6). 



