386 ILLUSTRATIONS OF THE DYNAMICAL THEORY OP GASES. 



The number of collisions between two particles of the first kind, s, being the 

 striking distance, is 



and for the second system it is 



2N* J* V2J8V. 



n _ 9/3 



The mean velocities in the two systems are ' and -. ; so that if Z, and l t 



JTT /TT 



be the mean distances travelled by particles of the first and second systems 

 between each collision, then 



PROP. X. To find the probability of a particle reaching a given distance 

 before striking any other. 



Let us suppose that the probability of a particle being stopped while 

 passing through a distance dx, is adx; that is, if N particles arrived at a 

 distance x, Nadx of them would be stopped before getting to a distance x + dx. 

 Putting this mathematically, 



or N=Ce-*. 



dx 



Putting N=l when x = 0, we find e'" for the probability of a particle not 

 striking another before it reaches a distance x. 



The mean distance travelled by each particle before striking is - = I. The 



probability of a particle reaching a distance = nl without being struck is e~ n . 

 (See a paper by M. Clausius, Philosophical Magazine, February 1859.) 



If all the particles are at rest but one, then the value of a is 



a = irs'N, 



where * is the distance between the centres at collision, and .2V is the number 

 of particles in unit of volume. If v be the velocity of the moving particle 

 relatively to the rest, then the number of collisions in unit of time will be 





