TRANSACTIONS OP THE SECTIONS. 31 



y/j V3' to V„ Vj ; and the circumstances of the encounter in the one case are pre- 

 cisely similar to those in the second. Hence, omitting for the sake of brevity the 

 quantities d^ &c., and cp, which have the same values in the two cases, we find 



A (ii, vv CO A ii.> '?.. Q =A (i\, n\, ro/. ii'., v\, Cd- • • • (5) 



If we now write 



log/(l,'?,C)=F(MV^;,m,«), (6) 



where /, m, n are the direction cosines of the velocity V of the molecule M, 

 taking the logarithm of both sides of equation (5), 



F,(MiV?/,/ni«,) + F,(M,V^/,m,M,)=F,(MiV?/V»V*'i) + F,(M,V=Z',m>',). (7) 



The only necessary relation between the variables before and after the encounter 

 is 



MiV5+M,V^=MiV'?+M,V^ (8) 



_ If the right-hand sides of the equations (7) and (8) are constant, the left-hand 

 sides will also be constant; and since /„ m^, ?«i are independent of l^,m,,n.^, we 

 must have 



Fi=-AMiV? andF,= -AM,V=, (9) 



where A is a quantity independent of the components of velocity, or 



/i(^»VuC.) = Cie-""^'''', (10) 



/.(^.,'7.,!r.)=C.e-^^-"'''' (11) 



This result as to the distribution of the velocities of the molecules at a given 

 place is independent of the action of finite forces on the molecules during their en- 

 counter ; for such forces do not affect the velocities during the infinitely short time 

 of the encounter. 



We may therefore write equation (1) 



d^=Ce~^'^''^^'''''^^^\lidrjdCdxdydz, (12) 



where C is a function of xyz, which may be different for different kinds of molecules, 

 while A is the same for every kind of molecule, though it may, for aught we know 

 as }'et, vary from one place to another. 



Let us now suppose that the kind of molecules under consideration are acted on 

 by a force whose potential is \/r. The variations of xyz arising from the motion of 

 the molecules during a time bt are 



o 



hx=^U,by=T)ht,bz=(U; (13) 



and those of |, r^, ( in the same time due to the action of the force, are 



fi^=-^6M'?=-ffiMC=-f8' 04) 



If we put 



'^ = logC, (15) 



''" d^d,dCdxdydz -='-'^^'^^'+^'+^'^- • • (1«) 



The variation of this quantity due to the variations 5.r,, Syi, (5.-,, 8|i, Sv,, Sf, is 



(«l+'|+4:)" 



