( 815 ) 
We put the expression for v in the form 
>) : 
: : 2 ZE 
It is easily seen that v is constant (=| ) as. long as no 
mn 
point is in collision with the walls. During the times that there 
are one or more collisions with the walls, v is much larger than 
Ze 
WZ as the third (positive) term surpasses the second (negative). 
me 
We suppose the time of collision to be very short. The path of the 
. 
representing point fvar during the collision remains finite (also if d 
@ 
approaches 0). The path of the representing point is such that if 
there are no collisions the p, are constant, while the ¢ are changing 
linearly with the time, while during a collision the representing 
point “springs” in a very short time to a new position, where all 
the q, remain the same and also the p, except those which correspond 
to the material point, which has suffered the collision. In both cases 
the trajectory can never cut itself, if it returns in the same point, 
it must be closed. 
We can distinguish two extreme cases: 1st. the collision lasts very 
short in conparison to the average time between two successive 
collisions and 2"¢. the reverse is the case. 
In the first case the velocity will, during intervals which are 
of the same order of magnitude as those between the collisions, 
2e 
have the value we — and it will largely deviate from this value 
mm 
during very short intervals. If we represent v as a function of the 
time by a graph, this will consist of pieces parallel to the axis of ¢ 
(during the intervals mentioned) interchanged by very steep tops of 
which the maximum ordinate depends on the maximum value of 
de N° . we 
i: During an interval of time, very long with respect to the 
average time between two successive collisions, the graph will show 
very many tops of various heights. In a sufficiently long interval 
the tops of each kind are likely to occur in each part of the path. 
9 
. . . . . aé& 
The time during which the velocity | X~ shall be the same 
mL 
fraction for every path that is sufficiently long, with deviations 
which are small in comparison with the quantities themselves. For 
53 
Proceedings Royal Acad. Amsterdam. Vol. XIII 
