( 400 ) 



velocities, and to ascertain whether the P integrated and siinuned 

 over these spaces becomes constant in conrsc of time. 



It stands to reason that here the |)r()l)lem of the (li.stril»Mtion of 

 velocities will be the simplest, becanse it is modified directly by tlie 

 impact, and the distribntion of place oidy indii-ectly. 



In agreement with § 2 where we considered an area from the 

 original extension lying between /„ and /„ -f- '^h •» order to examine 

 the placing, we shall take here a part from the extension determined 

 by limits of velocity lying infinitely near each otiier, bnt covering 

 a finite part of the 3/i-dimensional space of coonlinates. In connection 

 with the condition that ^v' = ^', we take from tlie .'^//-dimensional 

 space of velocities an element of a spherical shell, wliose radius 

 is ^^-2'ü^ To this corresponds a [)rismatic or cylindi-ical pari of the 

 extension, the base of whicii is represented i)y the element in (jiiestion. 

 With regard to the distribntion of velocities the points from these and 

 similar prismatic or cylindrical tubes come to the same thing. The 

 elements of the spherical shell i-epresent the projection of the tubes 

 on the space of velocities. Now it remains to investigate whether 

 the quantity of substance, which originally is found above the elenient 

 mentioned in a given tube, will not linally have spread uniformly over 

 all the tubes, so that the same quantity will be found ai)ove every 



element of the same size. If so, >Ss = 1 V'" A.'// V-'^/r will again l)ecoine 



mininnim, if dr represents the size of such an element, and P'<It 

 the quantity which is projected in <h. 



We may also call the points from the element of the spherical 

 shell the points of velocity of the systems, and the vector, which 

 joins the origin with such a point of velocity, I'epresents all the velo- 

 cities of the system both with regard to magnitude and to dii-ection ; 

 the projections of the ^■ector on the 'in axes of the space are the 

 components of the molecular velocities. 



The best way of setting forth the gradual uniform dispersion of 

 these points over the mentioned hypersphere is perhaps by availing 

 ourselves of Borel's mode of representation, and l)y partially following 

 his method. ^) 



BoKEL imagines that in the same 3/i-dimensional space in which 

 the coordinates of the molecules are laid out (so that we get in this way 

 a point representing the total arrangement for every system) also the 

 components of velocity are projected starting from the repj-esenting 

 point mentioned. The vector then obtained represents the Aclocity 



1) "Sur les Principes de la Theorie cinétique des gaz" par Emile Borel. Annates 

 de l'Ecole Normale Supérieure llle Série, 1906, N". 1, p. 9. 



