Illustration of the Theory of Gases, 



425 



degree to a want of preparation in the mind of readers, who 

 are confronted suddenly with ideas and processes of no 

 ordinary difficulty. For myself, at any rate, I may confess 

 that I have found great advantage from a more gradual 

 method of attack, in which effort is concentrated upon one 

 obstacle at a time. In order to bring out fundamental sta- 

 tistical questions, unencumbered with other difficulties, the 

 motion is here limited to one dimension, and in addition one 

 set of impinging bodies is supposed to be very small relatively 

 to the other. The simplification thus obtained in some 

 directions allows interesting extensions to be made in others. 

 Thus we shall be able to follow the whole process by which 

 the steady state is attained, when heavy masses originally at 

 rest are subjected to bombardment by projectiles fired upon 

 them indifferently from both sides. The case of pendulums, 

 or masses moored to fixed points by elastic attachments, is 

 also considered, and the stationary state attained under a one- 

 sided or a two-sided bombardment is directly calculated. 



Collision Formula. 



If u', v' be the velocities before collision, u, v after collision, 

 of two masses P, Q, we have by the equation of energy 



F(u ,2 -u 2 ) + Q{v' 2 -v r2 )=0, (1) 



and by the equation of momentum, 



~P{u'-u) + Q(v'-v) = 0. ..... (2) 



From (1) and (2) 



u +u=v' + v, (3) 



or, as it may be written, 



u — v' = v— u. 



signifying that the relative velocity of the two masses is re- 

 versed by the collision. From (2) and (8), 



(P + QK=(P-Q)i* + 2Q^ i 



As is evident from (1) and (2), we may in (4), if we please, 

 interchange the dashed and undashed letters. Thus from the 

 first of (4), 



(P + Q)tt=(P-Q)t*' + 2CK 



