of Perfectly Elastic Spheres. 23 
Solving this functional equation, we find 
ace, Ole ae 
If we make A positive, the number of particles will increase 
with the velocity, and we should find the whole number of par- 
ticles infinite. We therefore make A negative and equal to 
I s 
— 7a» 80 that the number between x and x+dr is 
2 
NCe™ 2 dx. 
Integrating from v= —a to z= +, we find the whole num- 
ber of particles, 
NC Vaa=N, ..C 
f(z) is therefore 
1 
aa 
Whence we may draw the following conclusions :— 
lst. The number of particles whose velocity, resolved in a cer- 
tain direction, lies between x and x+ dr is 
1 = 
N Teor wma tap weet ie tlt (1 
wie we o 
2nd. The number whose actual velocity lies between v and 
v+dbv is 
4 
08 oJ or 
3rd. To find the mean value of v, add the velocities of all the 
particles together and divide by the number of particles; the 
result is 
ve 
Drennan Net ones (2) 
N 
mean velocity ==. i Adah eee) 
V 7 
4th. To find the mean value of v*, add all the values together 
and divide by N, 
mean valiievof g*= Bars cn tio ty (4) 
This is greater than the square of the mean velocity, as it 
ought to be. 
It appears from this proposition that the velocities are distri- 
buted among the particles according to the same law as the 
errors are distributed among the observations in the theory of 
the “method of least squares.”” The velocities range from O to 
# , but the number of those having great velocities is compara- 
tively small. In addition to these velocities, which are in all 
directions equally, there may be a general motion of translation 
