320 Prof. L. Boltzmann on the Assumptions necessary 



is to be explained if we ascribe to Lesage's ultramundane 

 particles the properties of gaseous molecules ; for, whilst the 

 sun protects the earth from certain impacts, it reflects other 

 particles towards the earth which would not otherwise have 

 reached it. Only, if the particles are not reflected by the 

 earth and sun, but absorbed or perhaps reflected with a loss 

 of energy, would attraction be produced between the sun and 

 the earth. 



I have carried out calculations similar to the above in my 

 treatise, " Further Studies on the Thermal Equilibrium of 

 Gaseous Molecules/' * for a single kind of molecules after a 

 somewhat different method, which, I believe, is characterized 

 by great clearness ; and I venture to recommend to the 

 reader who is so disposed a perusal of the first section of this 

 treatise. 



We can easily apply the method there described to a mixture 

 of two kinds of gas. For the first of these let \/x . (f>(x, t)dx 

 be the number of molecules in the unit volume whose energy 

 at the time t lies between the limits x and x+dx. Let 

 cf>(x, i) have a similar meaning for the second kind of gas. I 

 have then used 



\ZxK<j>(Xj t)dx<j>(X, t)dXyjr(x, X, x')dx' 



to express the number of impacts which occur in unit time 

 and volume between two molecules of the first kind, so that 

 before impact their energies shall lie between the limits 

 x and x + dx, X and X + dX, whilst after impact that of the 

 one molecule lies between x and x' + dx'. 



By variable distribution of conditions we are always to 

 understand the number of impacts during a very short time 

 divided by that time; yjr has a similar meaning for the impacts 

 of the molecules of the first kind with those of the second 

 kind. 



Then we easily convince ourselves that 



^- Bflfo t) = j m ^ x+ * d xdx'[4(x', Oflj+X-*, t) 



<s/x'(x + X-x')ylr{x, x + X-x f , x) -<j>(x, t)<j>{X, t) y/xXf{x, X, a!) \ 

 + (j>{x', ^^(x + X—x 1 , t) x \/a!(x + X—a!)x{ri, x + X—x\ x) 

 -${x, *)3>(X, *) V^X%K X, x')] 



wherein we obtain \/X =nt — by interchange of <f> and <I>, 



* Sitzber. d. Wien. Akad. d. Wissensch. vol. lxvi., October 1872. 



