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generator according to which the tangent plane through PZ touches 
C2, If C? is constructed in such a way that the plane through 
PL and v, touches it, d cuts vin the point of contact with C?, 
We can thus make a geometric image of all directions of velocities 
namely in the following way: 
Given a ray va of the complex, cutting plane PRS in L; construct 
a cone C?, having P as vertex, touching the planes PQR and PQS 
according to PR and PS and touching moreover va. Construct the 
harmonic ray PL’; then the rays through L’ cutting PQ, also cut 
the complex rays through Z in the points of contact with C2 If 
we construct all the rays of the complex, a pencil of cones is 
formed; to each tangent plane PLva belongs a cone and a pencil 
of complex rays through a point of PL. 
So whilst to each tangent plane a cone belongs, two tangent 
planes Plv, PLv', belong to each cone; the ray d through !Z' 
cutting PQ and vg, also cuts v'g in a point of contact with C?. 
The planes dPQ, dRS, dvg, dv', form a harmonic pencil. 
12. Finally a few general observations may be in their place 
at the conclusion of this communication. 
a. It is clear, that if we consider the four vertices P,Q,R,S as 
the points of coincidence of two projective systems, each of these 
points plays the same part; by regarding, as was done in the 
beginning, the edges PQ and RS as conjugate polars of a focal 
system a limiting condition has been introduced. 
And the introduction of this condition is allowed as the principal 
tetraeder and one direction of a velocity do not determine, the position 
of the homologous points of two projective systems though they determine 
the complex of rays. By the second assumption, that of the conic K?, the 
focal system is determined. As it is possible to choose in three different 
ways a pair of edges as conjugate polars and moreover the point of 
intersection “4 can be assumed in two different planes, the point A 
can be determined in twelve different ways on a direction of velocity va. 
b. The number of solutions for the determination of the point A 
on the direction of velocity vq diminishes, when two of the vertices, 
say R,S are imaginary. So PQ and RS form the only possible pair 
of opposite edges. The point of intersection Z can be determined in 
two faces (PRS and QRS). If now also the points P and Q coincide 
as is the case for the motion of invariable systems, only one solution 
is possible. 
c. The entire preceding consideration is independent of the length 
of the velocities. It is also possible to find constructions for which 
use is made of that length. This will be done in a following com- 
munication. 
