( 589 ) 
2. We again start from the point of the motion of a system =, 
remaining congruent to itself; we construct the direction of velocity va 
of a point A and mark off on va a distance AA, representing the 
velocity of 4., When © moves in such a way that A follows the 
direction A Ay, then A comes to Ay ; however we can suppose the 
motion to be different but yet A reaching A„. 
3. For this purpose we construct the rectangular triangle A 4,4,, 
of which A;,A, is parallel. to the’ principal axis. / and A4, is 
normal to it. We then bring the plane &€ through AA, normal to/; 
let X be the point of intersection of § and /. If we now consider the 
points of = situated in & and if we compare the position of 4, to 
that of A, we then see that A and 4; can be regarded as homologous 
points of two similar systems in § The same reasoning can be 
applied to each plane normal to / brought in a similar way through 
points BB,,C C,...; from this ensues : 
The spacial systems ABC.., 4; Bz Cz are projective; the tetrahedron 
of coincidence has as opposite edges / and /,,; all points of J are 
points of coincidence, the two other points of coincidence are the 
cyclic points of any plane cutting the principal axis / at right angles. 
Each point of / is the pole of similitude of two similar plane 
systems situated in a plane normal to 4. . 
Now to thesystem 4, B, C, ..is communicated a translation parallel 
to the principal axis the amount of which is 4, A, ; in this motion this 
system remains congruent to itself; the tetrahedron of coincidence of the 
systems A; B, C,. ..,and A, B, C, has further as opposite edges 
land l,, but now ail points of /, are points of coincidence and 
the points of coincidence on / coincide in the point of intersection of 
band Ag. 
4. By the preceding considerations the connection between the 
systems = and =, has become evident. If the intermediate system 
of the points Ax B: Cr... . be denoted by =, , we then have the 
following relations : 
a. The systems 2 and ©, are projective; the points of coinci- 
dence are all points of / and the cyclic points on /,. 
b. The systems 2, and =, are projective; the points of coin- 
cidence are all points of /,, and the point of intersection of J and 
Ap counting double. 
c. The systems © and 2, are projective; the points of coin- 
cidence are the cyclic points on /,, and the point of intersection 
of 7 and A, counting double. 
39 
Proceedings Royal Acad, Amsterdam Vol, IV, 
