DR. PLtJCKER ON A NEW GEOMETRY OF SPACE. 
733 
within the plane of that direction and passing through the point {x\ y\ z'). By replacing 
in the last equation r and s by and j~ j, we obtain, in order to represent that 
plane, the following equation, 
(A-E^-^)+(B-D^(y-y)4-(lH-E^+Dy)( 2 -^)=0. ... (4) 
14. Again, this equation being, with regard to (x 1 , y\ z 1 ), of the first degree, proves 
that, conversely, all rays confined within a given plane meet in the same point of that 
plane. 
15. A complex the rays of which are distributed through infinite space in such a 
way that in each point there meet an infinite number of rays constituting a plane, and, 
conversely, that each plane contains an infinite number of rays meeting in the same 
point, may be called a linear complex of rays. We may say, too, that, with regard to the 
complex, points and planes of the infinite space correspond to each other ; each plane 
containing all rays which meet in the point placed within it, and each point being tra- 
versed by all rays which are confined within the plane passing through it. 
16. A linear complex of rays is represented by the linear equation (1), but it is easily 
seen that this equation is not the general equation of a linear complex. The following 
considerations lead us to generalize the preceding developments and to render them by 
generalizing more symmetrical. 
Hitherto we determined a ray by its two projections within XZ, YZ, 
x—rz- f § , 
y=sz+a, 
whence its third projection within XY is derived, 
ry—sx—ra—s% (5) 
This equation furnishes the new term (r<r—s§), which, like f and <r, depend upon r and s 
as well as upon a! and y' in a linear way. 
Again, from the equations 
= 0 , 
tg-\-uff-\-w= 0, 
expressing that the ray (r, s, f, <r) falls within the plane (t, u, v, to) represented by the 
equation 
tx-\-uy~\-vz-{-tv=0*, 
we deduce 
w v . . 
ys— t <s=(rc-so). ......... (6) 
* Henceforth We shall make use of four plane-coordinates t, u, v, w, and accordingly represent a point by a 
homogeneous equation. Sometimes, where symmetry and brevity require it, likewise x, y, z shall be replaced 
hy £/0, 17 / 0 , £/0. Accordingly, by introducing the four point-coordinates t, tj, (, 6, a plane is represented by 
a homogeneous equation. 
MDCCCLXV. ' 5 I 
