(590 ) 
So in space the rules for projectivity as found in the plane do 
not hold good unconditionally ; the system =, is not similar to =; 
for though in the sections normal to / two similar systems are 
situated, those which are parallel to 7 contain congruent systems ; 
as however 4,, is a face of the tetrahedron of coincidence and the 
condition of affinity is satisfied, the theorem ensues: The extremities 
of the velocities of the points of a spacial system © which remains 
in its motion congruent with itself, form a system =, affine with =. 
The tetrahedron of coincidence of = and =, agrees with that of = and 
the system =’ situated at infinitesimal distance, with which = will 
coincide in its motion in the direction of the velocities. 
5. As little as in the preceding paper is it necessary to give 
applications of all relations existing between two affine spacial sys- 
tems; now too we shall content ourselves with a single one. It is 
well kwown that the directions of velocity of the points of a right 
line are generatrices of a hyperbolical paraboloid; so the extremities 
of these velocities are also situated on this paraboloid and the 
systems © and +, being affine, they lie on a right line, on which 
they generate a range of points similar to that on the given 
right line. 
6. From the preceding ensues that with given direction of velo- 
city va we can assume a point 4, on it, representing the extremity 
of the velocity of a point 4. According to the preceding paper however 
A cannot be taken arbitrarily on va but is constructed on it as 
foot of the distance between e and vq; reversely the direction 4», 
when A is given, will be completely determined as tangent to the 
circle having AX as radius, whilst 4; A, must be parallel to J; so 
the direction of velocity 44, must be situated in a definite plane. 
7. We now pass to the consideration of the velocities of a 
system = changing projectively and we retain. the annotations 
used in the preceding communication. After a closer analysis the 
point A, homologous to 4 proves to have been found by first assu- 
ming / as axis of rotation and then by letting the points of = 
perform a motion round this axis, which we might call a “rotation 
of similitude”; after this the points of =, have performed a rotation 
round 4. With these successive motions the points of coincidence 
were retained. Now we again assume the tetrahedron of coincidence 
PQRS. And we construct, taking the found principles into consider- 
ation, the point .4, homologous to A. 
The motions now to be performed we call “projective rotations.” 
badk 
