DE. PLUCKEE ON A NEW GEOMETEY OE SPACE. 
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5. A right line of the second description, which we shall distinguish by the name of 
axis, is determined by any two of its points. We may select the intersection of the axis 
with the planes XZ and YZ as two such points, and represent them by the system of 
equations 
xt +z t v=l, ) 
yu-\-z u v~\, ) 
( 3 ) 
or by the following equally symmetrical, 
t =pv-\-zs, 
u—qy-\-7t. 
( 4 ) 
In making use of the first two equations, the four constants x, y, z t , z u are the coordi- 
nates of the axis, indicating the position of the two points within the planes XZ, YZ. 
In making use of the second system of equations, p, q, zs, k are the four coordinates 
of the axis, this axis being fixed by the intersections of two planes, one of which is the 
plane projecting it on XY, and determined by two of the four coordinates, 
t — l ?= -? U~7t—~ i 
x y 
while the other plane determined by the two remaining ones, 
t=pv=-~v, u=qv=—%, 
and represented by the equation 
px+qy+Z= 0, 
passes through the axis and the origin. 
6. If we consider the four coordinates of a ray as variable quantities, we may in 
attributing to them any given values successively obtain any ray whatever transversing 
space. But in admitting that an equation takes place between the four coordinates, 
rays are excluded : we say that the remaining rays constitute a complex represented hy 
the equation. 
In admitting two such equations existing simultaneously, those rays the coordinates 
of which satisfy both equations constitute a congruency represented hy the system of 
equations. A “ congruency” contains all congruent rays of two complexes, it may be 
regarded as their mutual intersection. If we admit that three equations are simul- 
taneously verified by the four coordinates, the corresponding rays constitute a configura- 
tion (Strahlengebilde, surface reglee) represented hy the system of three equations. A 
configuration may be regarded as the mutual intersection of three complexes, i. e. as 
the geometrical locus of congruent rays belonging to all three complexes. Four com- 
plexes or two configurations intersect each other in a limited number of rays. The 
number of rays constituting a configuration, a congruency, a complex, and space, are 
infinites of first, second, third, and fourth order. 
7. If rays are replaced by axes, complexes, congruencies, and configurations of rays 
are replaced by complexes, congruencies, and configurations of axes. 
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