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of the length of the velocities; however it is necessary to investigate 
more closely the above-mentioned tetraedral complex. 
3. When a system remaining congruent to itself is displaced, 
two opposite edges of the tetraeder of coincidence (principal tetraeder) 
are real: the principal axis / and the line /, at infinity common to 
the planes normal to /. Now there must be on / as well as on 
lo two real or imaginary vertices; if we call the first pair P, Q 
and the second &,S, then P,Q are the double points of two congruent 
ranges of points on / and therefore united in one point at infinity; 
k, S are the double points of two congruent systems in the plane 
hp at infinity, therefore two cyclic points of any plane normal to J. 
From this ensues : 
Of the four vertices of the principal tetraeder of the complex two 
coincide in A,,, the two other being the cyclic points of a plane nor- 
mal to J; of the faces of this tetraeder two likewise coincide inA,,. 
Suppose the direction of a velocity vg is given and we wish to 
construct the point A possessing this direction of velocity ; then we 
have to construct the line d of shortest distance from / and v,; this 
cuts v, in A. If we bring a plane « through d normal to vq then 
this plane is normal in A to the locus of 4; the rays through A 
in @ are the normals to the orbits and at the same time rays of a 
focal system having J and /,, as conjugate polars. Thus the focal 
system is connected with the tetraedral complex. 
4. Let us now suppose, that the system in motion changes projec- 
tively. If we imagine two positions of the system, then the points 
of coincidence are the vertices of the principal tetraeder PQRS, which 
tetraeder we suppose for our further consideration to be constructed 
and for the present entirely real. The tetraedral complex is determined 
by the principal tetraeder and the line connecting one pair of homo- 
logous points. This ray, however, is not only the line connecting 
two homologous points; the same complex appears, when it is regarded 
as the bearer of «? pairs of homologous points. 
If now this ray is the direction of velocity vq of a point A of 
this system, it is evident that for the determination of this point 
further conditions must be introduced, namely such as permit the 
construction of the point A as well as of the plane a. 
5. In the first place a pair of opposite edges of the tetraeder must 
be conjugate polars of a focal system; PQ and RS to be taken for 
these. ‘This condition, however, is not yet sufficient, as in the 
