316 Prof. L. Boltzmann on the Assumptions necessary 



therefore 



vy(o = v / y'(o f =z; . . . . . .(16) 



whence 



(v f y'oy = v' 2 -vy 2 -v 2 rfu 2 = \v(2gs<j- ys 2 o + ya 2 o) + ^gj- 



Therefore 



dx dy dz = 4Mv 2 v f ryy'o'dv dQ dO. 

 Since, further, 



x = v r2 , y={m + M)v /2 + 2Mv'rg' + Mr 2 , 2=?AyV, 



it follows that 



dx dy dz=^Mv ,z ry ,2 o'dv f dQ'dO f ; 



whence the relationship to be proved between dv dQ dO and 

 dv' dG 1 dO' follows. But, according to equation (15), it 

 follows from (14) that 



■J" 



F = $TT*8Wdv) \ ) \ V*V*f K V ! ,t)F(V',t)rTSCTdYdTdSdO.(n) 



n and p can be found from N and P by simple exchange of 

 the function F and the mass M for / and m, and in place of 

 8 we have \ the diameter of a molecule of the first kind. 



Hence 



n=8v*i?f(v,t)dv\*0\ I \ \ Y*f(V,t)rTs<TdYdTdSdO, 



J V ° 



p = 8TT*v)*dv\*0\ \ \ \ Vy(v',ty(V',t)rTS<rdYdTdSdO, 

 Jo Jo Jo Jo 

 mce 



?i) = 2ttX 2 f " f T C'ifA -//, )Y*rr S <rdV dT dS dO ) 



Jo Jo Jo Jo f J 



= > (19) 



+ 2tt8 2 i (fF'i-fFJYtrTsadVdTdSdO,) 



Jo Jo Jo Jo 



V ^=2wA«f"(''rf , '(FF l -FF 1 )*«rT«rdi,dTdS«iO 



*^o Jo Jo Jo 



and thence 





=- — i<iii l 111 * i — L x i) 1 ' rTS(ra,} axai3av\ 



+ 2*rS»| UW 1 -fV 1 )*rT*rdv<n!dBdO. ) 



«- o i o Jo J 



