740 
DR, PLUCKER ON A NEW GEOMETRY OF SPACE. 
first case 
in the second 
LL' . MM' 7 
NN' _ =*’ 
UU' . VV' 7 
U'V' ~ ^ 
all rays thus determined constitute the linear complex, represented by 
so — rff=k, 
the axis of which is OZ. 
If #=0, the linear complex is of a peculiar description, all its rays meet the same 
right line, the axis OZ. 
31. The results of [29] may be derived in a direct way. Let (rf, y', z') be any point 
of space; according to the general equation (10) its corresponding plane with regard to 
the complex 
will be represented by 
sq—rff—lc ... (24) 
y'oc—ot?y=.k(z—z!) (25) 
In putting tf=0, y'— 0, this equation shows that all planes corresponding to points of 
the axis of rotation OZ are perpendicular to this axis (in the case of oblique coordinates 
parallel to XY). 
If the point fall within XY, we get by putting z'=0, 
y'x—ody—kz\ 
consequently the corresponding plane passes through O. In denoting the angle which 
it makes with the axis of rotation by X, we obtain 
whence 
cos X— 
V y H + 
y'*-\-ri 2 =Jc i tan 2 k. 
(26) 
Hence we conclude, 
Light lines parallel to and at an equal distance from the axis of the complex are met 
under the same angle by planes corresponding to their points. 
32. The following results are immediately derived from (26). 
The plane jp corresponding to any given point P passes through OP, O being the 
projection of P on OZ. Let the plane jp and the right line OM perpendicular to it in 
O turn round the axis OZ, through an angle -, and denote them after turning by and 
OM'. The projection of OP on OM' is a constant, and equal to p. So is the perpen- 
dicular drawn from P to p’. 
Again, k being given we may, by determining X, construct the plane corresponding to 
a given point, and, conversely, by determining OP, construct the point corresponding to 
a given plane. 
