DR. PLUCKER ON A NEW GEOMETRY OE SPACE. 
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made use of by the equation (6), which indicates that all rays are parallel to a given 
plane. This plane, if drawn through the origin, is represented by the equation 
ax-\-by—z , 
obtained from (6) by writing -•> - instead of r, s. 
It may be sufficient here to state that a configuration of rays, if represented by 
three linear equations, in which the coordinates r, s, g, a are replaced by t, u, v x , v y , 
becomes a hyperboloid. 
9. A configuration of axes represented by three linear equations would be a para- 
boloid if the coordinates x, y , z t , z u were employed, but becomes a hyperboloid if these 
coordinates are replaced by p, q, ar, z. We shall here consider the last case only, and 
may for that purpose directly replace the equations (6)-(ll) by the following ones: — 
ap +bq = 1, (12) 
ca +d»= 1, (13) 
a'p +c'ar =1, . (14) 
Vq +d'x = 1 (15) 
a"p-\-d"z= 1, (16) 
b"q+d , B = 1 (17) 
Any three of these equations, involving six constants, are sufficient to determine the con- 
figuration. 
If, after having replaced^?, q, ar, z by 
_£?, _£^, i, I, 
x y x y 
we regard x, y, z t , z u as variable, (14) and (15) may be written thus, 
x=a!z-{-c', 
y=b'z+d\ 
representing within the planes XZ, YZ two right lines (AA, BB') which are the locus of 
points (A, B) where the axes of the configuration meet the two planes. 
In regarding vr and z as coordinates of a right line, the equation (13), being written 
thus, 
ct-\-du=l, 
represents a given point (E), 
x=c, y=d , 
enveloped within XY by the projections of axes. Therefore all axes of the configura- 
tion intersect a third right line (CC') parallel to OZ and meeting XY in E. 
Hence we conclude that the configuration represented by the three linear equations is 
a hyperboloid. Its axes meet three given lines, two of which, AA', BB', fall within 
XZ, YZ, while the third, CC', is parallel to OZ. 
