732 DE. PLUCKEE ON A NEW GEOMETEY OE SPACE. 
12. Again, let a congruency of axes be represented by the equations 
Atf+By +C z t -\-~Dz u +1=0, 
A'x + B'y + C % + B'z u + 1 = 0. 
By successively eliminating z u and z t we may replace these equations by the following 
two, 
ax-\-by -\-cz t +1 = 0, 
dx-\-b'y +<+ M +l=0, 
the six new constants of which are derived from the primitive constants. In regarding 
x, y, z h z u as point-coordinates (where z may be written instead of z t and z u ), the last 
equations represent two planes. The six coordinates of both planes, 
t=a, u=b, v=c, 
t=a', u=V , v=d, 
are the six constants of the congruency, consequently the congruency is determined by 
means of these two planes and the axes of coordinates. 
Suppose both planes to be known. Draw any right line meeting them in M and M7, 
project M on XZ and M' on YZ. The right line joining the two projections B and A 
is an axis of the congruency. 
If we project on>XZ and YZ any point of the right line JK along which both planes 
intersect each other, the right line joining both projections, B', A', is an axis parallel to 
XY. All axes obtained in that way meet, within XZ and YZ, both projections of JK. 
Hence the axes of the congruency parallel to XY constitute a paraboloid. The ray 
within XY is obtained by projecting the point where the traces of both planes meet on 
OX and OY and joining both projections, B" and A", by a right line, See. 
13. After these preliminary discussions we shall now proceed in a more systematic 
way, and henceforth exclusively make use of the coordinates r, s, g, <r. When a complex 
of rays is represented by the linear equation 
Ar+Bs+D<r+Eg + 1 = 0, (1) 
we may easily prove that the infinite number of rays passing through a given point of 
space are confined within the same plane, and, conversely, that the infinite number of 
rays confined within a given plane meet within the same point. 
In order to select among the rays of the complex those passing through a given point 
(d, y\ z '), the following two equations, 
a?=rz'+G , ] 
y=sz'+<r,/ 1 ’ 
are to be added to the equation of the complex. By eliminating g and <r we get 
(A-E2>+(B-DZ>+(l+IV+Dy)=0 (3) 
This equation being of the first degree with regard to the remaining variables r and s, 
shows that all corresponding rays are parallel to a given plane, and therefore confined 
