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RS, the latter in Z/; construct PL harmonically conjugate to PL' 
relatively to PR and PS; according to the preceding ZL must be 
determined on PL. When the complex of rays is determined the 
ray through A must cut not only PL, but also a ray in the plane 
QRS belonging to the pencil of rays with centre Q, projective 
to the pencil P/RSL; this ray QL" corresponds to PL. So we have 
to determine this ray QZ" according to the known projectivity of 
the pencils with the centres P and Q and to construct the line 
through A cutting PL and QL"; by this the plane a is at the 
same time known. 
6. Given «; the connection of the point of intersection of « 
and PQ with the point of intersection Z' of « and RS produces a 
focal ray. We determine farthermore the pole L of the line of 
intersection of @ and PRS relatively to A?, construct the ray QL" 
corresponding to the ray PL in the two projective pencils P/RSL, 
Q/RSL" and bring through JZ a right line, cutting QZ" and the 
constructed focal ray. This right line is the direction vq and its 
point of intersection with the focal ray is A. 
8. The preceding considerations point to a connection existing 
between the investigations of A. SCHOENFLIES “Geometrie der Be- 
wegung” pages 79—129 and those of L. BURMESTER “Kinematisch 
geometrische Untersuchung der gesetzmässig veränderlichen Systeme”, 
Zeitschrift für Mathematik und Physik, vol. 20, pag. 395—405. The 
former treats very completely of the constructions ensuing from the 
focal system and the tetraedral complex belonging to it, when the 
system remains congruent to itself during its motion; the latter 
assumes the projective variability of the moving systems, but does 
not make use of the focal system. 
It would not be difficult to give a more general form to most 
constructions appearing in the former consideration ; this would however 
give rise to unnecessary repetitions; so it will be sufficient if this 
is shown in a single example. 
9. To do so we take the construction corresponding to that of 
the characteristic of invariable systems; so the question is to deter- 
mine according to the foregoing principles in the plane @ the right 
line a containing the points the directions of velocities of which lie 
in «. For this a must be the line of intersection of two homologous 
planes of the two systems at infinitesimal distance of each other ; 
so if we think on va the point A to be determined and the point 
A' at an infinitesimal distance of it, and the planes @ and « to be 
