(235 ) 
first we determine the section (H?) of the hyperboloid with the plane 
at infinity, then we construct the chords of intersection of H? with 
the imaginary circle (C°) in that plane. If the pole of one of those 
chords of intersection in reference to C? falls in H?, the hyperboloid 
is rectangular. 
3. By this method the problem of space is transformed into a 
problem of the plane; in the further treatment, however, we come 
across the difficulty of an imaginary conic C?. For a better insight 
into the problem, we substitute for the present an arbitrary plane 
for the plane at infinity, a real conic for the imaginary circle and 
then the problem is formulated as follows: 
Given a conic K? and a pencil of conics with the also real base 
points 1, 2, 3, 4; to determine a conic L? of the pencil, for which 
the pole of a chord of intersection with A? lies on Z?. 
4. If L? is found, we can still make the following remark about 
the solution: Let Z? intersect the conic K? in the points £,, Lo, 
Lz, Las let Ly, be the pole of Z, Lg in reference to K? and let 
L? be brought through Z,., then according to a known theorem 
L? will also pass through the pole Ls, of the opposite chord 
L; Ly"). So the points 1, 2, 3, 4, Lig, Ls, lie on the same conic. 
After this remark we can pass to the construction of the locus of 
the poles, supposing that £° describes the whole pencil. 
5. Let 4? be a conic of the pencil (1234): it intersects K? in 
the four points A,, 4,, A3, A4. These four points will give rise 
to six common chords A,A,, A, Azo A, Au, Az Az, Ay Ay, Az Ay 
which correspond to six poles Aj... . A34. If A? is replaced suc- 
cessively by all the conics of the pencil, every new conic gives rise 
to four new points of intersection: on K? these quadruples form 
an involution of the fourth order. It is clear, that if A, is chosen 
arbitrarily and conic A® is constructed, 4,, A3, A, on K? are also 
determined, and that reciprocally when one of the last points, take 
Ag, is chosen, A,, A; and A, are also determined. The six lines joining 
the quadruples of points by two envelop a curve C, of the third class °). 
1) STEINER-SCHRÖTER: Theorie der Kegelschnitte, (‘Theory of Conies”), II, 3rd 
edition, p. 526, problem 90. 
2) R. Sturm, Die Gebilde ersten und zweiten Grades der Liniengeometrie, (“The 
figures of the first and the second order in the geometry of the straight line)”, I, 
p. 29. 
Miuinowsk1, Zur Theorie der kubischen und biquadratischen Involutionen, (“Theory 
of cubic and biquadratic involutions”) Zeitschrift f. Math, und Physik, 19, p. 212 ete. 
