382 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 



It appears from this proposition that the velocities are distributed 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 to oo, but the number of those having great velocities is com- 

 jwiratively small. In addition to these velocities, which are in all directions 

 equally, there may be a general motion of translation of the entire system of 

 particles which must be compounded with the motion of the particles relatively 

 to one another. We may call the one the motion of translation, and the other 

 the motion of agitation. 



PROP. V. Two systems of particles move each according to the law stated 

 in Prop. IV. ; to find the number of pairs of particles, one of each system, 

 whose relative velocity lies between given limits. 



Let there be N particles of the first system, and N' of the second, then 

 NN' is the whole number of such pairs. Let us consider the velocities in the 

 direction of x only ; then by Prop. IV. the number of the first kind, whose 

 velocities are between x and x + dx, is 



a -Jir 

 The number of the second kind, whose velocity is between x + y and x + y + dy, is 



_ 



N'- -i=e * dy, 

 P-J* 



where /8 is the value of a for the second system. 



The number of pairs which fulfil both conditions is 



-, l<* ( 



NN'--e ~P~' dxdy. 

 afiir 



Now x may have any value from oo to + oo consistently with the difference 

 of velocities being between y and y + dy; therefore integrating between these 

 limits, we find 



NN' - f ===- f =edy ......................... (5) 



' 



for the whole number of pairs whose difference of velocity lies between y and 

 y + dy. 



