( 493 ) 
constructed, then a proves to be the polar of va relatively to the 
focal system. 
According to (6) a passes through a point of PL, moreover a lies 
in «, so it has a point in common with the line of intersection of 
the planes @ and PRS which is the polar P of L relatively to A?; 
from this ensues: 
The polar a of va relatively to the focal system cuts the plane 
PRS in the pole of LL’ relatively to A? and lies on a hyperboloid 
of which PQ, RS and va are three generators. 
10. Up till now we have supposed the four vertices of the prin- 
cipal tetraeder to be real. The constructions, however, can still be 
performed if we assume that the two vertices RS are conjugate 
imaginary. The two edges PQ and RS namely remain real, also the 
planes PAS and QRS, the imaginary edges PR and RS are repre- 
sented as imaginary double rays of an elliptic involution of rays 
with the centre P; the construction of the conic AK? and of the 
polar p remains however possible and therefore also the remaining 
constructions of the complex of rays and of the focal system. 
If, however, the four vertices of the principal tetraeder are ima- 
ginary, then the construction can no longer be performed, because 
according to the preceding it ought to take place in a plane (PRS), 
which becomes imaginary itself. As the constructions treated of here 
will also be considered from another point of view, this case shall 
for the present remain unnoticed. 
11. In the theory of the motion of an invariable system we 
imagine cylinders of revolution to be described round the principal 
axis. If one of these cylinders is constructed, the velocities touching 
these cross the principal axis under the same angle, so that they 
are tangents to helices of definite inclination. Let us now find out 
the analogon of these cylinders in the motion of projectively changing 
systems and let us to do so return to the formerly (5) constructed 
rays PL and PL' which are harmonically conjugate with respect to 
PR and PS. 
We suppose furthermore a quadratic cone C? to be constructed, 
the vertex of which is P which touches the planes PQR and PQS 
of the tetraeder according to the edges PA and PS; then the planes 
PQL and PQL' are conjugate polar planes of C° If we now bring 
a tangent plane to C? through PL, this touches the cone according 
to a generator lying in the plane PQL'; from this ensues: 
The right line d through L’, cutting PQ and va, also cuts the 
