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DE. PLUCKEE ON A NEW GEOMETEY OE SPACE. 
8. A configuration of rays or axes, represented by three linear equations, is, according 
to the choice of coordinates, either a hyperboloid or a paraboloid. Let the three 
equations of a configuration of rays be 
A r +Bs +C +E f =0,1 
AV+B's+C'+D'<r+E'e=0,l (5) 
A"r +B"s+ C" + T>"<r + E" f = 0. J 
From these equations we derive by elimination six new ones, each containing two 
only of the four variables. Let them be 
ar =1, (6) 
eg +dff =1, (7) 
a'r+c'g =1, (8) 
Vs +& e=l, (9) 
a"r+d"<r= 1, (10) 
b"s+c"g = 1 (11) 
In order to represent the configuration, the three primitive equations (5) may be 
replaced by any three of the six new ones. 
The equation (7) may be written thus, 
cx-\-dy= 1, (7*) 
x and y replacing g and a. It represents a right line within XY, intersected by the 
rays of the configuration. 
The equations (8) and (9) represent within XZ, YZ two points enveloped by the 
projections of the rays of the configuration; consequently the rays themselves meet two 
right lines passing through these points, and being parallel to OY, OZ. From the 
equations (8) and (9) if written thus, 
we immediately derive 
c'x= 1, c'z=a\ 
d'y= 1, d!z—V , 
representing the two right lines. 
Thus by selecting in order to represent the configuration the three equations (7), (8), 
(9), and interpreting them geometrically, we have proved that all its rays intersect three 
fixed right lines, one of which falls within XY, while the two remaining ones are parallel 
to OY and OX. Hence these rays, meeting three right lines parallel to the same plane, 
constitute a hyperbolic paraboloid. 
In determining the paraboloid, we may replace any one of the three equations we 
