742 
DR. PLUCKER ON A NEW GEOMETRY OF SPACE. 
which, after eliminating successively y and x, may be replaced by the following ones : 
x! , y , -^y ,, =k[{^-x , )z-(x 1, z'-x , z , % 
In denoting the coordinates of the two conjugate lilies by 
r 0 , s 0 , § 0 , <7 0 , and r\ s°, g°, <r°, 
the following relations are immediately obtained : 
Whence 
and 
x"-x< 
r 0 — z ii-. 2 i ’ 
?0— 2 ,l_ s , 
r°= k 
r"z' — 7,'z" 
.0 7 . Z 
t ~- /C x"y'-x'y" 
y» S l-yl Z » 
ff 0 — z" — z' 5 
s°= k- 
-Jc 
■ y z 
y, o_ ,y o_go_ (r o_( g ogo- Wo) 
(5 0 ?0— ^>o)(5°f 0 — r°G°) =k\ 
Not any two conjugate right lines intersect each other; if congruent they belong to 
the complex. 
35. A linear complex depends upon five constants, four of which fix in space the 
position of its axis. In the case of the equations (23), this axis falling within an axis 
of coordinates, there remains only one constant. The position of the axis of the com- 
plex and its remaining constant may be determined by means of the five independent 
constants of the general equation (7). 
For that purpose we shall make use of the transformation of coordinates. If the 
axes of coordinates be changed, the coordinates of a ray change at the same time, and 
we get formulae analogous to the formulae in the case of ordinary coordinates, in order 
to express the coordinates of one system by means of the coordinates in the other. 
36. Let 
x=rz+<>, 
y=sz+( 7 
be the equations of a ray referred to the system of coordinates ( x , y, z ). If referred to 
another system ( x y\ z'), its coordinates will be replaced by new ones (r', s', q, </), but 
their equations retain the same shape, 
sWa'+g', 
y= 5 ' 2 'H-<7'. 
