1038 PROFESSOR TAIT ON THE 



As the value of this expression depends in no way on the length of the free path, it is 

 clear that the average energy of all the particles is greater than that of the free particles, 

 by an amount which increases rapidly as the length of the free path is diminished. 



APPENDIX. 



A. Coefficient of Restitution less than Unity. 



Let us form again the equations of § 19, assuming e to be the coefficient of restitu- 

 tion. We have 



so that 



p ( u '- u )=-T^" ) ( u -v)=-Q(v'-v) > 

 P(u'2- u 8)=-^-^(u~v)((2P+Ql-e)u+Q(l+e)v) 

 Q(v'«-v«) = ^|^- ) (u-v)(P(l+e)u + (2Q + P(l-e))v) . 



The whole energy lost in the collision is half the sum of these quantities, viz., 



PQ(l-e)* 

 » P + Q ^ u vj • 



With the help of the expressions in § 22, we find for the average changes of energy of 

 a P and of a Q, respectively, 



1 P(u^ - i?) = - 2 [|p l ^ 2 (2(P/t' - Q/0 + Q(l - e)(h + h)) 



The first term on the right is energy exchanged between the systems ; and, as in the 

 case of e= 1, it vanishes when the average energy per particle is the same in the two 

 systems. The second term (intrinsically negative for each system) is the energy lost, and is 

 always greater for the particles of smaller mass. The average energy lost per collision is 



PQ(l-e 2 ) /l 1\ 



2(P + Q)U" h &/ 



It is easy to make for this case an investigation like that of § 23. But we must 

 remember that there is loss of energy by the internal impacts of each system, which must 

 be taken into account in the formation of the differential equations. This is easily found 

 from the equations just written, by putting Q = P : — but the differential equations 

 become more complex than before, and do not seem to give any result of value. [Shortly 



