( 498 ) 

 collisions forces acl which only depend on the place of the particles 

 and not on (he velocities, we may apply the formula I dp\ . . ,dq n ~ 



tdp l ...dq n lo an ensemble of such pairs of molecules, the former 



representing the extension-in-phase after, the latter that before the 

 collision. In the case discussed by Bot/tzmann the masses of these 

 molecules arc /// and m, so that we get : 



dx' dy' '/:' m* </£,' '/»/ >/C dx\ dy\ dz\ m x <l$\ dr( x </£', = 



J 



r= f '/.'• dy dz m' </$ <h\ </£ d.r' d;/' dz' m x 3 dS,' <h t ' dC,' 



As we may consider the coordinates during the collision as inva- 

 riable, it follows from this that : 



Cd§ drf <K' <iz\ </>/, dg\ = i<J* d H d: d$ x d Hx d;. 



§ 3. However as has been referred to above, we may, without 

 assuming anything about the mechanism of the collision, prove the 

 property by means of the formulae for the final velocities with 

 elastic collision, making use of the functional determinant. Another 

 method is followed by Wind in his above-mentioned paper (the 

 second proof) and by Boltzmanx (vol. II p. 225 and 226); this 

 method differs in so far from the preceding one, that the changing 

 Of the variables takes place by parts (by means of the components 

 of velocity of the centre of gravity), which simplifies the calcu- 

 lation '). A third more geometrical method is given by Wind in his 

 first proof. This last method seems best adapted to me to convey 

 an idea of the significance of the principle of conservation of exten- 

 sion-in-phase in this special case. I shall, however, make free to 

 apply a modification which seems an abridgment to me, by also 

 making use of the functional determinant. So it might now also be 

 called a somewhat modified first method. 



In the first place I will call attention to the fact that with these 

 phenomena of collision it is necessary to compare infinitely small 

 volumes; if we, therefore, want to use the formula: 



Jdg' dr[ d% dg', dr^ d$' 1 = I — - 1 - -M dj; d)^ d$ d§ x dt ix d^ 

 J | d (| . . . 6) | 



l ) It seems to me that in this proof Boltzmann does not abide by what he 

 himself has observed before (§ 27 and § 28, vol. 11), viz. that the equality 

 of the differential products means that they may be substituted for each other in 

 integrals. The beginning of § 77 and the assumption of du dv dw, and dUdV 

 d W, as reciprocal elements of volume, is, in my opinion, inconsistent with this 



