DR. PLUCKEE ON A NEW GEOMETRY OF SPACE. 
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be such that 
tx-\-wy-\-vz-\-wm= 0, . . (1) 
which equation, if geometrically interpreted, indicates that each point ^ falls 
within each plane or, which is the same, that each plane passes 
through each point ( — , X, — ) . I called such coordinates “ associated plane- and point- 
coordinates”*, and here we shall make use of that denomination. By two couples of 
associated either plane- or point-coordinates, 
t u v f! u' v' 
— ’ — ’ — ’ — i’ — /’ 
WWW WWW 
x y z x' y' z' 
— , , — | 5 — —5 
UT '57 •07 ® W 
the same right line is determined. 
We may employ homogeneous instead of ordinary equationsf ; accordingly each group 
of three coordinates is replaced by a group of four : 
t , u, v, w, vl, v', uo\ 
0C, y, Z, nr, x', y, z', W ! . 
2. Both planes ( t , u, v, w) and (ff, u\ v', w'), represented in point-coordinates by the 
equations 
tx -\-uy -\-vz +wc7 =0, 
t'x -j-u r y-\-v'z-{-w ! vj=0, 
are arbitrarily chosen amongst those passing through the right line, and may be replaced 
by any two others, the equations of which have the form 
(t-\-yrf)x-\-(ur\- i*u')y4- (v-\-/jijv')z^-(w-\-[Jjw')vi ■ = 0 , 
where p denotes any arbitrary coefficient. But the position of the right line with 
regard to the axes of coordinates OX, OY, OZ is not characteristically connected 
with such a plane, except in the case where the plane itself has a peculiar relation to 
the axes. There are four such cases : the plane may either pass through the origin, or 
project the right line on the three planes of coordinates. Accordingly, in putting 
w+(jt,w'= 0, v-\-pv'=0, w+|tW=0, t-\-yjtf= 0, 
the last equation successively becomes 
(tw 1 — t'w)x-\-(uw' — u'w)y + (vw' — v'w)z = 0 , ^ 
(tv' —t'v )x-j-(uv' —u'v )y—(vw'—v'w)m =0, 
( tu ' —t'u)x—(uv' —u'v )z—(uw'—u'w)&= 0, 
— -( tu' —t'u)y—(tv ' —t'v )z—(tw'—t'w)m= 0. 
* Geometrie des Raumes, No. 5. 
t I first introduced homogeneous equatipns into analytical geometry, Ckeele’s Journal, v. p. 1, 1830. 
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