100 Prof. L. Boltzmann on some Questions 



which is to be integrated with reference to n, q, r from -co 

 to + go . Since we assume that the molecules are reflected at 

 the solid walls of the vessel like elastic spheres, those molecules 

 which strike against the walls of the vessel with certain values 

 of q, r will return from the wall with the same values of q 

 and r. The impinging molecules differ from the returning 

 molecules only in the sign of n. The function / therefore 

 remains unaltered if q and r remain unaltered and only n 

 changes its sign, whence it follows that in the integral (20a) 

 each two terms neutralize each other ; consequently this 

 integral itself has the value 0. So also of the three summa- 

 tions, of which, according to equation (7), the quantity 8 3 %cf> 

 consists, we may integrate the first with reference to £, the 

 second with reference to r), and the third with reference to £, 

 between the limits -co and + cc. But since for infinite 

 values of the velocities /must necessarily vanish, we see that 

 also S 3 2<£=0. 



If, at the beginning of the time, the gas were contained in 

 a finite space not enclosed by walls (the integration-space), 

 then the expression (19) would be negative, and numerically 

 equal to the number of the molecules which emerge from the 

 integration-space during the time St. But then the number of 

 these molecules would be exactly equal to the expression (20), 

 so that S x 2(£ + &>2</> would still be equal to zero. So also o\2<£ 

 would assume a negative value, and S 3 2</> an equal positive 

 value, if the velocities of the molecules at the beginning of 

 the time were included within finite limits. 



If we wish to prove the equation 



8 1 2^ + 8 2 2^> + 8 3 S0=O (21) 



directly, without employing integration with reference to one 

 of the coordinates or velocities, we may proceed thus : — Let us 

 imagine each point of the parallelepiped do moved forward in 

 space with the velocity-components £, v, £, and each point of 

 the parallelepiped dm with the velocity-components X, Y, Z ; 

 whereby the first points of course are reflected at the walls 

 exactly like the molecules, and the latter also must be corre- 

 spondingly altered. 



Since X, Y, Z are not functions of £, n, £, neither of 

 the parallelepipeds alters its size. Let the parallelepipeds 

 be denoted in their original position by do and dco, and in 

 their new position, which they have reached by the motion of 

 their points during the time Bt by do* and dco*. Let <f> be the 

 value of this function at the time t, when in it the values are 

 substituted for the variables which correspond to the middle 

 points of the parallelepipeds do and dco, whilst this function 



