318 Prof. L. Boltzmann on the Assumptions necessary 



f °° r 00 r* C% r 2n 



14 4 14 ^,V, T, 0, v',V, S)dvdYdTd$dO 

 Jo Jo Jo ^0 "o 



r 00 r 00 C v f ^ r 2n 



= 1 ¥(«', V, V, 0', v, V, T, 0, S) A dv dV dT dO dS ; 



Jo Jo vo Jo JO 



where 



A=v 2 V 2 T:i/ 2 V'V. 



Therefore the third line of the equation (22) is also equal to 

 f V f f ] Bn f(f F i -/' F iO *> 2 VVr 5 <r ^ rfV dT dS dO ; 



■/O Jo Jo Jo Jo 



also the fourth is equal to 



/-CO ^00 P* Cl £ 



^0 Jo Jo Jo Jo 



2 S^F/^Fi-/' Fi')w a V 2 rT«o- <to dV dT dS dO. 







For the sum of the third and fourth lines we find, by 

 taking the arithmetic mean of the expressions found now and 

 previously, 



|f " C fT 2 ^B^fF.'-fF^l-^^v^Y^rscrdvdYdJdSdO. 



Jo Jo Jo Jo Jo / *1 



The same transformations are to be applied to the first and 

 second lines of equation (22). But in the latter expressions, 

 moreover, the two impinging molecules play exactly the same 

 part ; so that we may here exchange the quantities which 

 refer to the first molecule for those which refer to the second, 

 and vice versa. If both molecules belong to the same order, 

 we have again, generally, 



f °° C f* f 2 f 2 \(v, V, T, 0, v', V, T', 0', S) dv dY dT d$ dO 



». Jo t/0 Jo Jo 



■- f " f " f * f 2 rV(V, v, T, 0, V, i/, T', 0', S) ^ dV dT dS dO. 



Jo Jo Jo Jo Jo 



Therefore the first line of equation (22) may also be written 



ffrfc 



Jo "o Jo Jo Jq 



xn Mf'fi -ffi) v*V*rT8<r dv dY dT dS dO. 



If we once more apply to this the transformation mentioned 

 on the previous page, we obtain a fourth value for the first 



