Prof. Clausius on the Dynamical Theory of Gases. 435 
mised continuation of Prof. Maxwell’s paper, I beg now to forward 
my reply. 
In my paper I consider the following question: a molecule 4 
of a gas moves with a certain velocity in a space which already 
contains many other molecules m, m,, mg,... 5 and in so doing 
occasionally strikes against and rebounds from the latter; 
required the number of collisions made by pin the unit of time, 
or what is equivalent, the magnitude of the mean length of path 
between two consecutive points of collision. In my solution I 
confined myself to the case where the molecule « moves, and the 
others m, m,,... remain at rest; but at the same time I asserted 
that in the case where the latter molecules also move with the 
same velocity as , the number of collisions increases in the ratio 
of 1:4. I did not prove this assertion, because for the object I 
then had in view it was not necessary to enter into such parti- 
culars. Since Prof. Maxwell, however, in his treatment of the 
same subject, arrives at the ratio 1: “2 instead of 1: 4, I feel 
myself called upon to prove the accuracy of my former statement. 
Let us first assume that « alone moves, whilst m, m,, Mg++» 
remain at rest; and let w be the velocity of », N the number of 
molecules at rest in the unit of space, and s the magnitude to 
which the distance between the centres of » and any other mo- 
lecule must be reduced before a collision can occur. The number 
of collisions during the unit of time will then be 
oms2N. 
If we now assume that the molecules m, m,, mg... also move, 
we must replace the actual velocity v by the relative velocities of 
the molecule with respect to the molecules m, m,, mq - ++; and 
since these relative velocities differ from each other, the arith- 
metical mean of all their values must be taken. Representing 
this mean by 7, the number of collisions will be 
r7s?N, 
and consequently the ratio of the number of collisions in the two 
cases will be v: 7. 
Thus far Prof. Maxwell and I agree, so that it will not be 
necessary to enter here into the demonstration of the above for- 
mul ; we differ only in the determination of the mean value 7, 
Let be the velocity of any molecule m, and 3 the angle between 
the direction of its motion and that of the molecule w; the rela- 
tive velocity between » and m will then be 
V u? + v?—2uv cos 5. 
When the molecules m, m,, m... all move with the same velo- 
