776 
DR. PLUCKER ON A NEW GEOMETRY OE SPACE. 
Any two of the four planes represented by these equations are sufficient to fix the posi- 
tion of the right line. They contain five constants, which by division may be reduced 
to four, the necessary number upon which the line depends. Besides the five constants 
in the two equations we meet a sixth one in both remaining equations. But the right 
line being determined by the former five, the sixth ought to be a function of them. The 
equation of condition, connecting the six constants, may, for instance, be obtained by 
adding the three last equations, after having multiplied the first of them by — (tv! — tv), 
the second by (tv' — t'v), and the third by — (uv 1 — u'v). Thus we obtain 
( tu'—t'u)(vw'—v'w)—(tv'—t'v)(uw'—u'w)-\-(uv'—u'v)(tw'—t , w)=0 . . . . (3) 
The following six constants, taken with an arbitrary sign, 
+(uv'—u'v), + (tv'—t'v), +(tu'—t'u), + (tw'—t'w), +(uw'—u'w), +(vw'— v'w), 
may be regarded as the six coordinates of the right line . 
3. In quite a similar manner, when in order to fix the position of the right line 
we replace the two planes by the two points ( x , y , z, m) and (ct, y', z ', m'), we get the 
following equations in plane coordinates, 
(xtx 1 — x'rn)t -{-(ysr 1 — y'xn)u-{-(z^' — z!vs)v =0, 
(xz 1 -x'z )t -\-(yz' —y'z )u-(zJ -z'u)w= 0, 
{xy l -othy )t —( yz ' —y'z )v -(yvs'-y'^w— 0, 
. — (xy' —x'y )u — (xz! —x'z )v — (xm 1 — x'&)w=0, 
representing four points, the first of which is at an infinite distance on the right line of 
which the position is to be determined, while the three others are the points in which 
that line meets the three planes of coordinates. Accordingly we may likewise regard 
the six constants of the last four equations, taken with an arbitrary sign, 
-\-(xts ’ — x'&), + (y&' — yV), +(2 ot' — z'&), ±(yz' — y'z), ^(xz 1 — x'z), -{-(xy' — oty), 
as the six coordinates of the right line. These six coordinates are connected by the 
following equation of condition: 
(xy'-ody)(zv’—z'v ! )—(xz'-cJz)(yv'-y'™)+(yz'-tfz)(xv'—riT S )=<). . . (5) 
4. In denoting the distance of the right line from the origin of coordinates by \ the 
angles with it makes with the three axes OX, OY, OZ by a, (3, y, and the angles which 
the normal to the plane passing through it and the origin makes with the same axes 
by X, g>, v, the following relations are obtained : 
I. (uv 1 —u'v ) : — (tv 1 —t'v ) : (tv! —t'u) : ( tw' — Hw ) : (uw'—v!w) : (W— v'w) 
II. —(xtz'—x'ts) : (yJ—y'm) : (zn'—z'n) : (yz 1 —y'z) : —(xz! —x’z) : (xy'—ody) 
III. = cos a : cos/3 : cosy : hcosX : ticosy, : Scosv. 
5. Hence we conclude that 
cos (3, cos y, § cos x, 5 cos p, c> cos v 
cos a, 
