98 Prof. L. Boltzmann on some Questions 



where 



f=A*, y , z, ?, v', ?, 0, /I'-foy* h «-f, ■-«/, «>-?, 0* 

 dd=d?d v 'dtl. 



In order to express eta/ in terms of r/w, we must imagine 

 the magnitude and position of the centre of gravity as weJJ as 

 the direction of the line of centres at the moment of impact 

 completely fixed. On the other hand, the velocity- components 

 of the first of the impinging molecules may vary between tlie 

 limits (2). Jn the accompanying figure, in which the rect- 

 angle ££' CO does not necessarily lie in the same plane as the 

 triangle 0%u, let 0£ be the velocity of 

 the first molecule before impact, Ow 

 the velocity of the centre of gravity of 

 both molecules, and £C the direction 

 of the line of centres at the moment 

 of impact. The straight lines (not 

 drawn in the figure) 0%', OC, and OC' 

 therefore represent respectively the 

 velocity of the first molecule after 

 impact, that of the second before, 



and that of the second after impact. We have now to keep 

 the points and u, and the direction of £C unchanged, 

 whereas the point f? must describe the whole interior of the 

 parallelepiped dot. Since here the figure (■£' CC is always a 

 rectangle, and u always remains its centre, we see at once that 

 the point £' describes a parallelepiped da>' congruent with the 

 parallelepiped da, which represents nothing else than the 

 product of the differentials d% drf d£' ; consequently dco = d(o'. 

 (I have given a much more complete account of this trans- 

 formation of coordinates and its relationship to various for- 

 mulae of the kinetic theory of gases in my paper " On the 

 Assumptions necessary for the Proof of Avogadro's Law."* 

 H. A. Lorentz attains the greatest simplicity by using the 

 velocity-components of the centre of gravity instead of the 

 components of the relative velocity which 1 have employed 

 on page 641.) We therefore obtain from formula (15), 



dn! '= o* f f j'V cos do da dp d\$t; . . . (16) 

 and from formula (14), 



A = (</>' + (/>!'-</> - faXffi -J '/i'K'V cos do dco dp d\ St. (17) 



In order to find S 4 2<£ from this, we have to integrate this 



expression with reference to all differentials denoted by (/over 



* Boltzmann, Wien. Sitzungsher. vol. xciv. p. 037 (1880) ; Phil. Mag. 

 [5] vol. xxiii. p. 805. 



