28 Prof. Maxwell on the Motions and Collisions 
Putting N=1 when x=0, we find e~* for the probability of a 
particle not striking another before it reaches a distance z. 
The mean distance travelled by each particle before striking is 
—=/. The probability of a particle reaching a distance = nl 
without being struck is e~”. (See a paper by M. Clausius, Phi- 
losophical Magazine, February 1859.) 
If all the particles are at rest but one, then the value of a is 
a=s*N, 
where s is the distance between the centres at collision, and N 
is the number of particles in unit of volume. If v be the velo- 
city of the moving particle relatively to the rest, then the num- 
ber of collisions in unit of time will be 
otrs?N ; 
and if v, be the actual velocity, then the number will be 7; 
therefore 
a= as?N : 
vy 
where 7, is the actual velocity of the striking particle, and » its 
velocity relatively to those it strikes. If v. be the actual velocity 
of the other particles, then v= Vv,?+v,”. If v,=v,, then 
v= V2n,, and ef 
as /27s°N. 
Note.—M. Clausius makes «=47s?N. 
Prop. XI. In a mixture of particles of two different kinds, to 
find the mean path of each particle. 
Let there be N, of the first, and N, of the second in unit of 
volume. [et s, be the distance of centres for a collision between 
two particles of the first set, s, for the second set, and s! for col- 
lision between one of each kind. Let v, and v, be the coefficients 
of velocity, M, M, the mass of each particle. 
The probability of a particle M, not being struck till after 
reaching a distance #, by another particle of the same kind is 
e- V2rs2Nya 
The probability of not being struck by a particle of the other 
kind in the same distance is 
e: V4 no Nae 
Therefore the probability of not being struck by any particle 
before reaching a distance w is 
on™ V252N,+ VJ 140 57Ny) a 
5) 
