384 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASE3. 



It appears therefore that the quantity Pp* Qq* is diminished at every impact 

 in the same ratio, so that after many impacts it will vanish, and then 



q_ O_ 



Now the mean vis viva is f Pa* = Pp* for P, and ' - Qq* for Q ; and it is 



o o 



manifest that these quantities will be equal when Pp' = Qq*. 



If any number of different kinds of particles, having masses P, Q, R and 

 velocities p, q, r respectively, move in the same vessel, then after many impacts 



= Rr>, &c ............................... (7). 



PROP. VII. A particle moves with velocity r relatively to a number of 

 particles of which there are N in unit of volume ; to find the number of these 

 which it approaches within a distance s in unit of time. 



If we describe a tubular surface of which the axis is the path of the 

 particle, and the radius the distance s, the content of this surface generated 

 in unit of time will be wrs?, and the number of particles included in it will be 



Nirrs* ....................................... (8), 



which is the number of particles to which the moving particle approaches within 

 a distance s. 



PROP. VIII. A particle moves with velocity v in a system moving according 

 to the law of Prop. IV.; to find the number of particles which have a velocity 

 relative to the moving particle between r and r + dr. 



Let u be the actual velocity of a particle of the system, v that of the 

 original particle, and r their relative velocity, and 6 the angle between v and r, 

 then 



tt* = i? + ?* 2 vr cos 6. 



If we suppose, as in Prop. IV., all the particles to start from the origin at 

 once, then after unit of time the "density" or number of particles to unit of 

 volume at distance u will be 



1 -"' 



N- -e * 

 " " 



a. IT'- 



