DR. PLtiCKER ON A NEW GEOMETRY OF SPACE. 
749 
rays of one generation of a hyperboloid , while the given right line AB and its two con- 
jugate A'B', A"B" are rays of its other generation. In replacing O and Q! by other 
complexes arbitrarily taken among the complexes (58), the conjugate will be replaced by 
others, all intersected by the rays of the congruency starting from AB. Hence 
The right- lines conjugate to a given one , with regard to all complexes intersecting one 
another along a linear congruency , belong to one generation of a hyperboloid , while the 
right lines of its second generation are rays of the congruency meeting the given line. 
48. If a point move along a given right line of space, according to the last number, 
its corresponding ray generally describes a hyperboloid. We may say that the same 
hyperboloid is described by the ray which corresponds to a plane passing through the 
given right line and turning round it. If the ray be the same in both cases, the point 
where it meets the given line AB is a point of the surface, and the plane confining both 
AB and the ray, the tangent plane in that point. 
49. The hyperboloid generated by a ray of a linear congruency, the corresponding 
point of which moves along AB, varies if this line turn round one of its points C. All 
the new hyperboloids contain the ray which corresponds to C, but there is no other ray 
common to any two of them. If AB describe a plane, by turning round C through an 
angle sr, there will be one ray of a hyperboloid passing through any point of space. A 
linear congruency therefore may be generated by a variable hyperboloid turning round 
one of its rays. 
In an analogous way, a linear complex may be generated by a revolving variable con- 
gruency. 
50. While in each of the two complexes O and O' there is a fixed line — the axis of the 
complex around which its rays are symmetrically distributed — there is in a linear con- 
gruency a characteristic section parallel to both axes of the complexes, and a characteristic 
direction perpendicular to it. 
The characteristic section, if conducted through the origin O, may be represented by 
the equation 
ax -\rby cz— 0. 
The two right lines starting from O and parallel to the two axes of the complexes are 
represented by the double equations, 
x y z 
D = E _ T 
P__y__P_ 
D' — E ,— F‘ 
These lines being confined within the section, we get in order to determine the con- 
stants of its equation, 
aD +JE +<F =0, 
«D'-t-6E'+cF=0, 
whence 
MDCCCLXV. 
(D'E— E'D)&+(D'F— F'D)c=0, 
(D'E— E'D)a— (E'F— -F'E)<?=0. 
5 L 
