( 806.) 
PoiscarÉ and Zermero have elucidated that the trajectories of the 
systems are in general closed and there are many cases in which 
we find closed trajectories which do certainly not pass through all 
the points of the space #,-1. I shall give some examples as an 
illustration of this. If the kinetie energy which we have adopted 
using the dynamica! equations in the canonical form, is a homogeneous 
quadratic function of the momenta (the coefficients being functions 
of the coordinates), the system in which the momenta have been 
reversed, will be represented in the same space /2,-1- A certain path 
L being given, we can attain another possible path L’ by reversing 
the momenta at all points a,6,c¢ of the path 4; and the path Lf 
can be passed through in such a way that the interval of time between 
the two moments at which the positions 6’ and a’ are reached is 
equal to that between the moments at which first @ and 6 were 
attained. I shall term those systems reversed systems, their paths 
reversed. paths. It is now not necessary that the path and the reversed 
path are parts of the same trajectory and if they are not, there exist 
at least two totally insulated trajectories in the space /2,—1 and each 
of them cannot possibly run through all the points of 42,1. In 
order to give a simple example, we might consider the following 
case; within a sphere two material points are moving with equal 
velocity, which are reflected mutually and through the walls as 
perfectly elastic bodies. We can choose of all possible motions that 
one in which the two points move along the sides of a square. Now 
two assumptions are possible : 
1. The points move in the same direction; in that case the reversed 
path will never be reached. 
2. The points move in opposite direction; the path and the reversed 
path will be the same. 
Placing more points into the sphere we can, even if we ascribe 
a finite extension to the points, so that the distribution of velocities 
may be changed by the mutual impacts, always find an initial state 
such that in one case the path and the reversed are the same and 
in the other case not, while all the trajectories lie in the same space 
Fn) *). 
One might ask if in case the path of a system does not exactly 
reach each point of the space Zo, 1, it could not be possible that in 
the course of a sufficiently long time it would pass as near each 
point as we want to? In simple systems having a certain regularity 
this is impossible; what will be the case with complicated systems 
1) Compare Kervin Baltimore lectures p. 486. 
