DE. PLUCKEE ON A NEW GrEOMETEY OF SPACE. 
731 
11. In order to represent a congruency of rays, we shall here make use of the coor- 
dinates t, u, v x , v y . Let 
At +B u +0^ +Dy y +1=0, 
At + B'm + C'v x + D'Vy +1 = 0 
be its two equations. By successively eliminating each coordinate, we get four equations 
of the following form, 
at -\-bu ~{-cv x +1 = 0, 
dt -\-b'u -\-dv y +1=0, 
a"t-\-c'v x +d'Vy +1=0, 
b"u + dv x -\- d"Vy +1=0, 
any two of which involving six constants may replace the two primitive equations, the 
remaining two being derived from them. 
The first two of these equations, if t , u, v x and t, u, v y be considered as plane coordi- 
nates, represent two points (U, V) the coordinates of which are 
x=a, y=b, z—c , . (U) 
x=za', y—b\ z=d, (V) 
Consequently the six constants upon which the congruency depends, if referred to the 
three axes of coordinates OX, OY, OZ, are determined by means of the two points U 
and Y. Hence is derived the following construction of rays of the congruency. 
Trace through the two points U, V any two planes which intersect each other along a 
right line confined in the plane XY, and meeting OX, OY in the points D, F. Let 
E, G be the points where the two planes meet OZ. We shall get within the planes 
XZ, YZ the projections of a ray of the congruency by drawing DE, FG. The ray (AC) 
thereby completely determined will intersect the plane XY in the point C, the coordi- 
nates of which are 
x=]=OD, y=l= OF. 
If a plane be traced passing simultaneously through both points U, V, both intersec- 
tions E, G falling into one point A', the corresponding ray of the congruency A'C' 
intersects OZ. If the right line UV be projected on YZ, XZ, the projections meet OZ 
in two points A", A!". In these points OZ is intersected by the rays of the congruency 
parallel to OX, OY. The ray parallel to OZ is obtained by the point C" where it meets 
XY. The coordinates of C" are 
x=OB", Y=OF", 
D" and F" being the points where the projection of UV intersects OX and OY. 
Thus occurs to us the construction of rays passing through any point of OZ and any 
point of XY. We cannot go further into detail here. 
