( 591 ) 
8. We bring through 4 and RS a plane cutting the planes 
PQR and PQS in Pk and PS; we construct the conic C,° passing 
through A and touching Pf and P,S in the points R and S. The 
tangent through A to Cj? indicates the direction, in which A must 
move when the system of points in the plane PES performs a pro- 
jective rotation ; if we assume a point Ax on this direction of velo- 
city, the length of the velocity is determined. We have now to 
prove that the points of =, lying in the plane P,RS as well as 
other points are determined. 
9. The points of = in the plane P,RS can be instantly found 
by making use of two principles holding good for the projective 
rotation in a plane and which we may suppose to be known: 
a. The extremities of the velocities of all points of C,? also lie 
on a conic, touching Pf and PS in R and S; a point A; is 
sufficient to determine this conic. 
b. The extremities of the velocities of all points of a right line 
through P; also lie on a right line through P,. 
If we assume a point B outside the plane P| RS, we construct the 
PQ in Py. With the projective rotation the axis PQ being in all 
plane BRS cutting its points a line of coincidence, a projective 
rotation is also made in P,RS, having P as pole and where we can 
imagine the points to move in the direction of tangents to conics, 
touching Pol and PS in R and S. So we can determine the direction 
of velocity of B as tangent to the conic constructed with the aid 
of B. So of each point in the plane PRS the direction of velocity 
is situated in this same plane and according to the preceding perfectly 
determined. 
10. The relations just found are sufficient to determine to the points 
MBCT on her cireenons Ob -velociiy” An, BEE CC, al wes i 
but of the system of points Zr itself only 4; is assumed; B, , Co... 
have not yet been constructed. For these the following principles 
can be maintained: 
a. To be constructed the directions of the velocities of the points 
of the right line AB; these form a system of generators of a hyper- 
boloid, to the second system of which AB and RS belong; the 
right line through 4, and belonging with the two last named to 
one and the same system contains the points B,. . 5 
b. The generator of the system >, homologous to a right line 
a of =, cutting AS, is situated together with a and RS in one plane. 
ce. The generator of the system =; homologous to a right line 
b, cutting PQ, cuts PQ in the same point. 
39* 
