( 407 ) 



In my opinion the correct principle that the calculation of the 

 extension occupied by the combinations of the points of velocity 

 after the collision when that before the collision is known and vice 

 versa, would come to the same thing- as a transition to other vari- 

 ables in an integration, has not boon applied in exactly the correct 

 way. The property in question says that in an integral with transi- 

 tion from the variables SVSëWiin to (j^SS^S, the product of the 



differentials d^dtydSttg'^dg', may be replaced by ^^-~— 



d$dridSd$ } d7i 1 d£ 1 , if we integrate every time with respect to the corre- 

 sponding regions, but these expressions are not equal for <tll that. 

 The first expression may be said to represent the elementary volume 

 in the space of 6 dimensions, bounded with regard to § . . . S\ , the 

 second the elementary volume bounded with regard to £ . • • 5, 1 ). 



We have a simple example when in the space of three dimen- 

 sions we replace Ipdxdydz, which e.g. represents the weight of a 



body, by Ipr* sin i)<lrJd<hf , which represents the same thing, without 



<le dy dz having to be equal to r % sin & dr d& d<p. 

 So we have here : 



Cd§ dr( d$' dg, dn\ dC,\ = ( \ d j: s ~ ']] di d n d$ dg, ,h h <£,, 

 J J I d (§ — SJ 



which two expresssions represent the "extension" in the space of 6 dimen- 

 sions after the collision. That before the collision is ld$drid$dg 1 dTi l d$ i i 



so that, when the determinant = 1, the extension remains un- 

 changed by the collision. This proves really to be the case, as 

 Jeans shows. We may, however also consider this property as a 

 special case of the theorem of Liouville, and derive it from this '■'). 

 This theorem says, that with an ensemble of identical, mutually inde- 

 pendent, mechanic systems, to which Hamilton's equations of motion 



apply, \dp v ..dq n = ldI\...dQ n , when p x >--qn represent the coordinates 



and momenta of the systems at an arbitrary point, of time, P x . . . Q n 

 those at the beginning. Gibbs calls this law: the principle of conser- 

 vation of extension-in-phase, which extension we must now think 

 extended over a space of 2// dimensions. When now the two collid- 

 ing molecules are considered as a system which does not experience 

 any influence of other systems, and it is assumed that during the 



b Cf. Lorentz, 1. c. Abhandlung VII. 



2 ) As Boltzmann cursorily remarks: volume II p. '225. 



