DR. PLtJCKER ON A NEW GEOMETRY OE SPACE. 
785 
III. On a new System of Coordinates. 
23. We have hitherto determined the position of a right line in space in making use 
of the ordinary system of three axes OX, OY, OZ intersecting each other. The new 
question is whether we may substitute for this system another, by means of which we 
are enabled to fix immediately the position of a right line without recurring to points 
and planes. 
In the ordinary system of coordinates, (1) the position of a point is determined by 
means of three planes parallel to the planes of coordinates and meeting in that point, 
(2) the position of a plane by a linear equation between the three coordinates of a point, 
regarded as variable ; both point and plane depending upon three constants. 
In an analogous way a right line is determined by the intersection of four linear 
complexes. Such a linear complex depends upon the position of its axis and a con- 
stant (paper presented, No. 29). A right line, regarded as the direction of a force , 
belongs to the complex, if the moment of rotation of the force with regard to the axis, 
divided by its projection on the axis, be equal to the constant. Accordingly any four 
axes in space being given, the position of a right line is fixed by means of four constants, 
obtained by dividing the four moments of rotation with regard to the four axes by the 
four corresponding projections on the same axes. 
The four axes of the complexes constitute the new system of coordinates ; the four 
constants are the four coordinates of the given right line. The right line intersecting 
the four axes is the origin of coordinates, its four coordinates being equal to zero. 
In the new system of coordinates a right line is determined in the most general way 
by its four coordinates ; but an equation between the four coordinates is not in a general 
way sufficient to represent a linear complex, depending as it does on five constants. 
We may ad libitum increase the number of coordinates of a right line. 
24. Let P, Q, R, S, T, U . . be the axes of any number of complexes, and p, q, r,s,t,u.. 
the corresponding coordinates of a given right line (according to the last number). Let 
Q P = U p —p=0, = q=0, £l=* r -r=0, 
£l s = a s — s=0, £=0, Q M =E„ — u—0... 
be the equations of the complexes. In order to express that the complexes meet along 
the same line, the following equations of condition are obtained, 
only seen the papers, I hasten to mention it now. But, besides the coincidence referred to, the leading views 
of Professor Cayley’s paper and mine have nothing in common. On this occasion I may state that the prin- 
ciples upon which my paper is based were advanced by me, nearly twenty years ago (Geometry of Space, 
No. 258), but this had entirely escaped from my memory when I recurred to Geometry some time since. 
