( 816 ) 
example the number of tops of a certain kind be NV, for unity 
of time» then we shall find deviations of the order of magnitude 
of yN,, if we compare those numbers for different intervals equal 
to the unity of time. If now the duration of the collisions and also 
(e.g. by enlarging 7) the average time between successive collisions 
approaches 0, but in such an way that the first is infinitely small 
compared with the second, then the (v-2) graph will show an infi- 
nity of maxima and minima in a finite region. 
In the second extreme case a large number of points differing 
widely in phase, will always be in collision with the walls. If the. 
number of collisions of the given phase is .V pro unity of time then 
deviations of the order W/N (positive as well as negative) will occur. 
The length of the path through which the system passes on the line 
L in a unity of time will be the same for the majority of such- 
like intervals. The square root of the mean square of the deviations 
is small in comparison with the length of the path itself. 
Let us next consider a system in which ” perfectly rigid and 
elastic spheres of diameter o are enclosed in a volume J’. We fix 
our attention on a line-ensemble. The points of the line represent 
the phases of the system. In some of them the number of particles 
for each of the % equal elements J”, into which the volume V may 
u 
be divided will be exactly — =v; in others there can exist devia- 
tions whieh T shali indicate by +, for the element V,. The numbers 
ke 
Tr, answer to the condition Xr,=0. | shall assume the elements 
| 
V, great in comparison with the mean length of free path (of the 
molecules). Then a distribution with certain values of the numbers 
tT, will last for some time. We can therefore take rather long parts 
of the path 4 so that on each of them the value of 7, may be 
considered to remain constant. Let / be a part in which no deviations 
occur, / another with the deviations 7,. 
We have to determine 
n ( de ) 
zt bed 
f og, 
on those parts of ZL. 
The sum will show irregular deviations from one moment to 
another, caused by the accidental variations of the number of col- 
lisions. The mean value hewever can casily be denoted, if is different 
for 7 and /. The contribution to the value will depend for each 
element on the number of collisions occurring in the unity of time. 
