( 499 ) 



we must take infinitesimals of th« 2 n1 - order. We can, however, also 

 proceed in a somewhat different way. For how is the above formula 

 derived? By making use of the fact, that with a volume d$ di\ d$ d§, 

 d}^ l d^ l in the region of the §.. 5 X corresponds a volume 



rf| rf?i d$ d$ x dt] x d$ x 

 d (§ . . .6,) 



in the region of the §' . . . g ls or also that the first mentioned exten- 

 sion, occupied by the representing points in the space of 6 dimensions 

 before the collision, will give rise to the second extension after the 

 collision. We can, therefore very well compare these expressions 

 inter se, without integration, if only the second expression is not 

 interchanged with d§ drf d$ ig*, dr\ x dg v i. e. the volume element obtained 

 by dividing the extension after the collision in another way. 



We now suppose the points of velocity before the collision to be 

 situated in two cylindres, the axes of which are parallel to the 

 normal of collision. The bases of the cylindres are dOdO x and the 

 heights d<f and dff x . The extension occupied by the combinations of 

 the points of velocity is evidently equal to the product of the con- 

 tents of the cylindres: dOdO x dódó x , In case of collision the compo- 

 nents of the velocities perpendicular to the normal remain unchanged, 

 so the points of velocity are shifted in the cylindres in the direction 



of the axis, so that d becomes d', and d x becomes 6\. Between 



iese quantities e 

 m x <f x -f- ra(2d — d x ) 



me 4- m x (2^ — d) 



these quantities exist the relations : d = and d , 



m -\- m x 



, when m and m x denote the masses of the colliding 



m -j- m 1 



molecules (i.e. the same relations as between the normal initial and 

 final velocities with elastic collision. 



If we now wish to calculate the extension after impact we maj 

 make use of the fact that dO and dO x have not changed, so that 

 we need only examine what happens to dódó x or what extension in 

 the region of the ö'ó\ corresponds to the extension dddJ^ in the 

 region of the <fd x . 



According to the above this is: — - — -\ ddd<f x , and as it follows 



! d (d(f x ) | 



from the formulae for d' and ó\ that the absolute value of the 



determinant = 1, the extensions before and after impact are equal. 



The extension after the collision is, however, not equal lo the 



product of the cylindres in which the points of velocity will be found 



after the collision. This will be easily seen with the aid of the 



geometrical representation given by Wind. The extension before 



impact may be thought as the product of the extension in the space 



