778 
DE. PLUCKER ON A NEW GEOMETRY OE SPACE. 
p, s, ( — <r), §, and 7i will be the Jive absolute coordinates of the right line. The last 
of the four equations (2), representing the planes projecting the right line on the planes 
XZ and YZ, as well as the projections themselves, may now be written thus, 
x= rz-\-g, 
y=sz+a, 
r and s being the trigonometrical tangents of the angles made by the two projections 
with the axis OZ, % and <r the segments intercepted by them on the axes OX and OY, 
Again, let us divide the first five terms of the set of ratios II. by the sixth {pcj — x'y). 
In putting 
xvr — xts yts' — y’z? zw — zts y 
I r —— ~~~ 7 T — 7T, ~ ~1 / ' — C? 
xy' — ary ’ xy — xy xy — xy ^ 
yz ' — y'z xz' — x'z 
xy 1 — x'y xy 1 — x'y & 
where, according to the equation of condition (5), 
2>, q, ( — «), 7r, and ^ will be the Jive new coordinates. We meet four of them in the 
last two of the four equations (4), representing the two points where the planes XZ and 
YZ are intersected by the right line. These equations assume the following form, 
t =pv-\-7 rw, 
u=qv-\-xw , 
and may, in denoting the coordinates of the points within their planes by x y , z y , and y x , z x , 
be written thus, 
(v y t+z 9 v+w= 0, 
y x u+%jv ’±w= 0 ; 
whence 
We may add to the former six sets of equal ratios the two following: 
VII. = r : 8 : 1 (—a) : q(==,r<r—,sg). 
VIII. : 7 r : XJ=(px — q-rr) : jy : q : 1. 
10. We have thus obtained eight different systems of line-coordinates, the coordinates 
being the six terms of each of the eight sets of equal ratios I. to VIII. In changing the 
position of the origin and the direction of the axes of coordinates, the coordinates of 
each system are changed. But I do not here transcribe the formulae of transformation 
of line-coordinates, observing only that these formulae may be immediately transferred 
from one system to any other. 
