304 Mr. S. H. Burbury. On the Application [Feb. 7, 



1894) reasons for assuming as the law of distribution of velocities 

 among n spheres the expression 



in which the coefficients have yet to be determined. 



14, 15. The " a " coefficients must be all positive, the " b " coeffi- 

 cients all negative ; and the b coefficients express the fact that the 

 pairs of velocities to which they relate are not independent ; and the 

 fr's, being negative, express the fact that the two velocities are more 

 likely to be of the same than of opposite signs, so that there will be 

 on the average of any group of contiguous spheres a greater common 

 or stream motion than there would be were the velocities all inde- 

 pendent. 



16. The coefficients b must generally diminish as the distance be- 

 tween the two spheres to which they relate increases, becoming 

 evanescent when that distance is great enough. 



17. If the chance for a group of n spheres be of the form Ce~~*Q, 

 and for a group of n 1 spheres, part of the n spheres, Ce~ A Q"- 1 , Q M 

 and Q,_i must be connected by the relation 



If we effect the integration for one variable we find, if 

 Q w = ttl V -f b n uMt + a 2 w 2 2 + &c. 



in which 2a\ 2a, 



2a n 



Z Cl n 



This shows that as n diminishes the a coefficients diminish, and 

 since every b coefficient is negative the 6's increase in absolute value, 

 so that the ratios bb'/a or b z /a increase. On the other hand, as n 

 increases the o-'s increase, and the squares and products of the form 

 6 2 or bb' diminish. Whence it is inferred that as n increases the 

 function 



Q = 



tends to assume a limiting form. This limiting form must be T when 

 K = 0, and must be such as to make T r less than it would be were all 

 the velocities independent. It may then be assumed to be T + /cT r . 



