ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 385 



From this we have to deduce the number of particles in a shell whose centre 

 is at distance v, radius = r, and thickness = dr, 



-I &-) (r+y 



If -4= - {e~~^-e~~}dr .......................... (9), 



which is the number required. 



COR. It is evident that if we integrate this expression from r = to 

 / = oo , we ought to get the whole number of particles = N, whence the following 

 mathematical result, 



,00 (-ay (x+af 



dx.x(e~ ' -e~ ) = J-naa ..................... (10). 



Jo 



, 



PROP. IX. Two sets of particles move as in Prop. V.; to find the number 

 of pairs which approach within a distance s in unit of time. 



The number of the second kind which have a velocity between v and v + dv is 



The number of the first kind whose velocity relative to these is between r 

 and r + dr is 



N-= - (e f - 

 a^ir v v 



and the number of pairs which approach within distance s in unit of time is 



nn'-jrrs', 



4 _ (-* _(*) 



= KIT s'r've * {e a ' e * } dr dv. 



By the last proposition we are able to integrate with respect to v, and get 







(<*'+')* 



Integrating this again from r = to r = oo , 



(11) 



is the number of collisions in unit of time which take place in unit of volume 



between particles of different kinds, s being the distance of centres at collision. 



VOL. r. 49 



