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Mathematics. — “The elementary motion of space S,’’, by 
Dr. 8. L. van Oss. (Communicated by Prof. JAN DE VRIES). 
In this paper the writer, starting from the wellknown properties 
of the elementary motion in 83; and from a principle already 
formerly applied in his dissertation, intends to effectuate in a purely 
geometrical way the reduction of the elementary motion in S4 to 
a simultaneous rotation about two (perfectly) normal planes, or, as 
we shall call it, to a “double rotation”. 
THEOREM I. 9 and U' being two congruent systems in S4, 9| can 
be made to coincide with %' by two successive simple rotations. 
Let A and A’ be a pair of homologous S; in U and ’, intersecting 
in the plane c=/3'; then, if we determine the planes @’ and 9, the 
lines of intersection @//? and a@'//’ are a pair of homologous rays 
lying in the plane e=/', and consequently have a centre of rotation 
R in this plane. If now through A we bring the normal plane 
eg to a@, a rotation about @ will cause U to obtain a double line d, 
with ' (dj = a'//’'). This rotation being effectuated, we choose dj 
as the common axis of two homologous pencils of planes in 4 and 9’ 
and bring through an arbitrary point in this axis a S3 cutting 
it normally. This $3; being homologous to itself, intersects the pencils 
of planes in two homologous pencils of rays, which being congruent 
and possessing a double point, also have a double ray dy. So the 
rotation about eg has caused the plane (dj dj) to become a double 
plane of U and 4. A following rotation about this double plane 
brings U into coincidence with U. 
Corollary. The elementary motion in S, can be represented by 
a simultaneous rotation about two planes having a point in common. 
TreoREM II. A rotation in S, about a plane can be resolved 
only in one way into two simple rotations about planes one of which 
lies in a given S the other passing through a given point or being 
normal to this Ss. 
Definition. A rotation about a plane being resolved into two 
components the planes of which are respeetively in and normal to 
a given S3, the rotation of this S3 caused by the normal component 
shall be called the section of the given rotation with the Ss. 
TurorEM III. If two systems of rotation in S, are equivalent, 
then also their sections with any Sz are equivalent. 
Turorem IV. If the sections with an arbitrary S3 of two systems 
