( 491 ) 
preceding problem d was not only normal to / but also to v,. The 
plane /d constructed according to the supposition made there and 
a plane through / parallel to v, are normal to each other and are 
thus conjugate in respect to the planes through / and the cyclic 
points in the plane normal to !; by applying this last named 
principle to the case of projectively altering systems, we obtain the 
following construction for the point A, whose direction of velocity is va. 
Suppose the given direction of velocity va cuts the plane PRS in 
the point Z; construct in this plane the ray PL' harmonically 
conjugate to PL with respect to PR and PS, which cuts RS in ZL’; 
bring through Z' a ray cutting PQ and va; then this ray will cut 
va in the point A possessing the given direction of velocity. 
For the determination of the focal system the construction of the 
ray Z'A is not sufficient, as we know of the focal plane a belonging 
to A only that it passes through Z'A. To determine « entirely we 
have to notice that with the motion of congruent systems a cuts 
the plane 4, in the polar of the point of intersection Z of vg and 
Ap in respect to the imaginary circle in 4,,. This circle passes 
through the imaginary vertices in A, of the principal tetraeder 
and the point of intersection of / and 4, is the centre of it. By 
applying these principles to the case of systems changing projectively, 
we obtain the following construction. 
We assume a conic A’, touching PR and PS in R and S; we 
construct the polar p of Z through Z’ in reference to K? and we 
bring the plane @ through A and p; now @ is the focal plane of 
A, So with the motion of systems projectively varying the complex 
of rays and the focal system are connected with each other. 
6. The edges PQ and RS determine with vg a hyperboioid H2, 
on which the polar of va in reference to the focal system is also 
situated. This cuts the plane PRS besides in RS also in PL; so 
from this ensues that the polar of vq relatively to the focal system 
cuts the plane PRS in a point of PL. It is then easy to see, that 
PL is the polar of L' relatively to A’. 
7. Not until the inverse problems are solved, are the con- 
structions complete, thus (a) when for each point the direction of 
velocity and the focal plane are constructed, (6) when for each 
plane the focus and the direction of velocity of this focus are con- 
structed. We suppose in these constructions the complex of rays 
to be determined and A? moreover constructed. 
a. Given point A. Draw through A the line cutting PQ aal 
