1891.] 



On the Collision of Elastic Bodies. 



179 



altered except by collisions, provided /log/ and F log F become zero 

 when any of the velocities becomes infinite. 



14. The rate at which H approaches its minimum is found in the 

 case of two sets of elastic spheres of masses M and m, whose numbers 

 in unit volume are N and n, as follows : Let H = HX + K, where H t 

 is the minimum to which H tends. K is defined to be the disturbance 



, JT7" 



and the rate of subsidence. 

 K dt 



Suppose that the number in unit volume of spheres M having 

 velocities between U and CJ-HcZTJ towards the element of volume 

 U 2 dU sin a. doe. dp is, 



e -*M<i+D)u U 2 dU sin * d* 



in which 7&(l + D) is written for h in the usual expression for that 

 number. 



Similarly for the m spheres, h(l + d) shall be written for Ji. It 

 is assumed that the total energy is not altered by the disturbance, 

 which requires that 



l+D i+d 

 D and d are supposed so small that D 3 and d 3 are to be neglected. 



Then we find K = f - 



and 



dK 

 dt 



where s is the sum of the radii of M and m. 



Fence 1 ^^ 32 N -/(Mm) "** 



and K ^ K.Q G~ C . 



is proportional to the density and to the square root of the 

 absolute temperature. 



