Mathematics. — “On the Motion of a Fixed System”. By Prof. 
W. van per Woupr. (Communicated by Prof. J. CARDINAAL.) 
(Communicated at the meeting of September 25, 1920). 
§ 1. In the discussion of the motion of a plane system the atten- 
tion is usually directed to the locus of the points which at a given 
moment describe a point of inflexion of their paths and to the locus 
of the lines which in their motion at that moment touch their enve- 
lope at a cusp; these loci are indicated as the inflexional circle and 
the cuspidal circle. The starting point is here the so-called formula 
of Savary for the radius of curvature of the path of a point, resp. 
for the radius of curvature of the envelope of a curve (or a straight 
line) of the movable system. 
Such a discussion of the singularities in the motion of a fixed 
system in space is not very simple, the expressions for the curva- 
ture and the tortuosity of the path of a point are such that they 
do not invite further conclusions. As far as I know these singulari- 
ties are only dealt with in the well known book of ScHoENFLins ‘) ; 
he there draws attention to the remarkable relation between the 
points A of the movable system and the points A’ of fixed space 
when to each point A the point A’ is conjugated which is the 
centre of the sphere of curvature of A in its path, and to the fact 
that in the “inverse motion” A is the centre of the sphere of cur- 
vature of A’ in its path. 
I wish to reach these results in an entirely different way; I shall 
make use of the so-called method of the movable system of axes 
(triedre mobile), for the application of which to kinematics we can 
refer to the text-book of Koenies °). In the $$ 2, 3 I shall therefore 
repeat a few well known formulae. 
§ 2. By 7r(0,, X,, Y;,Z,) and’ 7, (0, X, Y,Z) we understand 
two equally orientated right angled systems of axes which move 
relatively to each other. The velocity relative to 7 of a point 
1) Dr. A. ScHornriies: Geometrie der Bewegung in synthetischer Darstellung. 
(Leipzig, Teubner, 1886). 
2) G. Koenis: Legons de Cinématique (Paris, HERMANN, 1897). 
38 
Proceedings Royal Acad. Amsterdam. Vol. XXIII. 
