(592) 
11. Let us now suppose the point of =; to be determined; we 
let this system also rotate projectively, allowing P, Q and all points 
of RS to be points of coincidence. If we now bring, in a corresponding 
way as was done when & was made to pass into =, , planes through 
PQ cutting RS in A), Ry. ... , we arrive at the construction of 
=, and we can easily state rules for the connection between =, and 
=, corresponding to those for = and =, ; at the same time the 
general theorem ensues: 
The extremities of the velocities of the points of a spacial system =, 
changing projectively in its motion, form a system >, projective 
to 2. The tetrahedron of coincidence of = and =, agrees with that 
of = and the system =’ at infinitesimal distance from it, with 
which = will coincide in its motion in the direction of the velocities 1). 
12. The construction of A, belonging to A remains possible when 
the points P and Q as well as & and S are conjugate imaginary, 
We suppose on PQ as well as on RS an elliptic involution of points 
to be given of which P,Q and R,S are to be the imaginary double 
points. Now we can construct as in the entirely real tetrahedron of 
coincidence a plane ARS cutting PQ in PL). 
In this plane the conic G,? is determined by the following data: 
the point A; pole and polar P, and RS; the condition that each 
pair of rays of the involution P,/RS is conjugate polar with respect 
to C°. So the system =, as well as =, can be constructed. 
13. The comparison of the constructions of this communication with 
those of the preceding one finally gives rise to some observations. 
a. The focal system formerly applied does not appear in this 
paper; it has been replaced by other assumptions. This focal system 
served as a help for the determination of the directions of velocity 
of spacial systems at infinitesimal distance from each other; we now 
see that the length of the velocities could be used for that purpose. 
The binding conditions for the construction of the directions of velocity 
rest on the identity of the tetrahedron of coincidence between = and 
=, and that between = and the system =" lying at infinitesimal 
distance, 
ee 
1) lt is to be noticed that for acceleretions theorems can be found corresponding 
to those now treated. For similar and affine plane systems BURMESTER did so in 
his “Kinematik”. For spacial systems remaining congruent Dr. P. ZEEMAN Gzn. gave 
a deduction in Problem 182 of the ‘“Wisk. opgaven van het Wisk. Genootschap” 
Vol VIIL with statement of literature. Note the last already quoted paper of BuRMESTER. 
