ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 



383 



This expression, which is of the same form with (1) if we put NN' for 

 N, a' + f? for a 2 , and y for x, shews that the distribution of relative velocities 

 is regulated by the same law as that of the velocities themselves, and that 

 the mean relative velocity is the square root of the sum of the squares of the 

 mean velocities of the two systems. 



Since the direction of motion of every particle in one of the systems may 

 be reversed without changing the distribution of velocities, it follows that the 

 velocities compounded of the velocities of two particles, one in each system, are 

 distributed according to the same formula (5) as the relative velocities. 



PROP. VI. Two systems of particles move in the same vessel ; to prove 

 that the mean vis viva of each particle will become the same in the two 

 systems. 



Let P be the mass of each particle of the first system, Q that of each 

 particle of the second. Let p, q be the mean veloci- 

 ties in the two systems before impact, and let p', q' 

 be the mean velocities after one impact. Let OA = p 

 and OB = q, and let AOB be a right angle; then, by 

 Prop. V., AB will be the mean relative velocity, OG will 

 be the mean velocity of the centre of gravity ; and drawing 

 aGb at right angles to OG, and making aG = AG and 

 bG = BG, then Oa will be the mean velocity of P after 

 impact, compounded of OG and Ga, and Ob will be that of Q after impact. 



Now 



P+Q 



therefore 



and 



and 





P+Q 



(6)- 



