1888 .] 
Prof. Tait on the Mean Free Path. 
225 
4. On the Mean Free Path, and the average Number of 
Collisions per particle per second in a Group of 
Equal Spheres. By Prof. Tait. 
There is general agreement as to the value of the quantity 
which expresses the fraction of the whole group whose members 
have speeds from v to v + dv; and as to the corresponding Mean 
Path The “mean,” in this case, is found accord.ing to the 
definition given by De Morgan: — 
“ The arithmetical mean, or average, is always to be 
understood when the word mean is mentioned, unless the contrary 
be specified.” 
Thus, using the word in its proper sense, the mean speed is 
the mean time of describing a free path is '^(n^p^/v); 
and thus also the Mean Free Path is %{n^p^). 
A quite different thing is the Mean of the Free Paths described 
by one particle in T seconds, or by T particles in one second 
(which, in a perfectly communistic group of S.lO^o per cubic inch, 
is of course the same). Yet the term Mean Free Path is usually 
applied to this quantity. 
The matter is of no consequence whatever in investigations con- 
nected with the Kinetic Theory of Gases, because in these the 
distribution of speeds has to be taken account of, and thus 
(about which all are agreed) alone comes in. 
The value of this wrongly named mean is easily found. During 
nJT: a particle has speed from v to v + dv, and its mean path is 
p^. Let be the number of such paths, i.e. the number of 
collisions it had after describing such a path. Then, if be the 
space it described under these conditions, we have 
= nJY.v = Q^p^ . 
Taking means (properly so called) we get 
S/T = :S(SJ/T = ^(?^^?;), the mean speed of a particle during T 
seconds. This is the same as the mean speed of all, at any instant. 
C/T = 2(C„)/T = the mean number of collisions per 
particle per second. 
