1420 
operation ') by AA. ScrHornruies*) and his predecessors only consider 
such A’s that an application of them to a point A produces a point 
B, the coordinates of which are found from those of A by a linear 
orthogonal substitution. By these operations the distance between 
two points therefore does not change. 
It seems natural to introduce for space-time symmetry-operations 
too the restriction that an application of them to two four-dimensio- 
nal xyz-ict-points (or rather 2'x?2*x*-points) A and B does not change 
the four-dimensional distance A B. 
In the first place we thus limit our considerations to those A’s 
the algebraic representation of which is a linear orthogonal four 
dimensional substitution and to corresponding symmetry-elements. 
Secondly, (as was also done by ScHorNFrims and his predecessors 
in an analogous sense) we exclude those A’s, the repeated applica- 
tion of which to a point A gives an infinite number of points at 
the same time within a finite space or within a finite time-interval 
at the same place. 
Thirdly it will prove desirable to introduce still one restriction, 
which has not its analogue in the three-dimensional problem. 
From the algebraic substitution mentioned we see, that 2", 2’, v° of 
the new point B depend on z of the original point A. Thus, applica- 
tion of the A’s in question to A gives a point B that is displaced 
in the course of time to an infinite distance even when A remains 
on the spot. This fact is an objection against the consideration of 
such a A; an objection however that may be avoided by considering 
only the final result of subsequent applications of more than one 4 
of the kind mentioned to a point A, of a AA therefore. 
So we limit ourselves to the consideration of such A’s, the appli- 
cation of which to a point A gives a point B with a world-line 
parallel with the «*-axis, when the world-line of A has that direction. 
In the next §§ we shall see to which kind of A’s we are led by 
this restriction. 
§ 3. Geometrical meaning and analytical indication of one of the 
kinds of operations considered. In a R, with coordinates 2, 2, 2° 
and at=vict, R, be an arbitrary linear space of three dimensions. 
We shall call R, a symmetry-space’) (symbol r) when the 
corresponding operation (symbol X, name space-time-reflection) changes 
1) In the same sense as f.i. a rotatory-reflection is a AA. 
2) A. Scuoenriies. Krystallsysteme und Krystallstructur, Leipzig 1891. 
5) This name has already been used by P. H. Scuoute, Verh. Kon. Ak. Amst. 
Eerste Sectie Il 7 (1894) p. 16. 
