113, 114] System of Loaded Spheres 101 



There is an exactly similar equation for {c /2 }, and by addition of these two 

 equations we obtain, after simplification 



_ O r 2 



( C 2 + C '2J = C 2 + C 'S + 4 r 2 (y + ^'2) _ j^ JJ2. 



If we use the symbol A to denote the increase caused by collision of the 

 quantity to which it is applied, the equation just obtained may be written in 

 the form 



. 



{A(c 2 +c' 2 )} = r 2 4 (OT2 + OT ' 2) _ ............... (242)) 



( <> K ) 



in which U has again been replaced by V. 



114. So long as we remain in ignorance of the law of distribution of 

 coordinates, it is impossible to carry any further the process of direct 

 averaging. We may, however, proceed as follows. 



An expression has just been found for the mean change of translational 

 energy produced by collisions in which the molecules have given velocities. 

 A summation extending over all collisions which occur during a short interval 

 of time dt will lead to an expression for the total change in the translational 

 energy of the gas during that period. From the form of equation (242) it is 

 clear that this expression will contain r 2 as a factor. But as we are neglect- 

 ing powers of r of a degree higher than the second, we may put r = in all 

 terms multiplied by r 2 . It follows that to find 2A (c 2 + c' 2 ) as far as r 2 , we 

 need only calculate 2 (cr 2 + -or' 2 ) and 2 U z on the assumption that r = 0, the 

 summation extending in each case to all the collisions which occur in the 

 time dt. 



This neglects a correction which is required by the considerations men- 

 tioned in 109, a correction which would consist in adding to the expression 

 for 2A (c 2 -f- c' 2 ), a quantity equal to the sum of all the separate corrections 

 to be applied to the right-hand member of equation (242) in cases in which 

 this equation fails. Now each separate correction will clearly contain r 2 as a 

 factor, and on averaging this must be further multiplied by a factor propor- 

 tional to the number of cases in which the correction is required ; i.e. by 

 a factor which vanishes when r = 0. The correction to the final result is 

 therefore of a higher order of small quantities than r 2 , and may therefore 

 be neglected. 



For a similar reason we may, in calculating 2 (-or 2 + r' 2 ) and S U 2 , 

 assume the law of distribution of velocities to be that which would obtain in 

 the case of r = 0. We therefore assume the number of molecules per unit 

 volume for which c and vr lie within ranges dc, dtff to be 



........................ (243). 



