complex becomes 
DR. PLUCKER ON A NEW G-EOMETRY OE SPACE. 739 
D<r+Eg+F(sg — r<r)+C =0, j 
D<r+Eg-}-F(sg — r<r)+Bs=0, >•••••••• (23) 
D<7+Eg+F(sg — r<r)-|-Ar=0. j 
28. In order to represent a linear complex by equations of the utmost simplicity, let 
us take any plane XY, XZ, YZ perpendicular to the characteristic direction, and draw 
through its corresponding point O the axis OZ, OY, OX. The resulting equations will 
assume the following forms, 
F(s§— r<r)-\-C =0, ) 
‘ B s +E ? =0, 1 (23*) 
A r +D<r=0. J 
The planes corresponding to all points of a right line having the characteristic 
direction are parallel to each other ; and conversely the locus of points correspond- 
ing to parallel planes is a right line of that direction. Hence we conclude that there 
is one fixed line, the points of which correspond to planes which are perpendicular to it. 
Consequently, on the supposition of rectangular coordinates, we may in only one way 
represent a linear complex by means of equations assuming the form of those above. 
Q 
29. In order, for instance, to get the first of these equations, which by replacing — - 
by k may be written thus, 
sg — r<r=k, 
it will be sufficient to direct OZ along the fixed line. As no supposition is made either 
with regard to the position of the origin on OZ, or to the direction of OX and OY 
within the plane XY which is perpendicular to OZ, this equation will remain abso- 
lutely the same if the system of coordinates be moved parallel to itself along OZ, or 
turned round it. In other terms, 
A linear complex of rays invariably remains the same if it be moved parallel to itself 
along a fixed right line or turned round it. 
The fixed right line may be called the axis of rotation , or merely the axis of the 
complex. 
30. We may give different geometrical interpretations to the last three equations, 
involving each a characteristic property of a linear complex of rays. 
Any two planes XZ, YZ intersecting each other along OZ being given, rays of space 
may be determined either by their projections on both planes, or by the points where 
they meet them. In the first case, if a third plane intersecting XZ, YZ along OX, 
OY at right angles be drawn, there are two planes LMN, L'M'N', parallel to each other, 
passing through the two projections LN, M'N, and meeting OZ, OY, OX in N and N', 
M and M', L and L'. In the second supposition, denote the two points of intersec- 
tion by U and Y, and their projections by U' and V'. Accordingly U'U, V'V, and U'V' 
maybe regarded as the projections of UY on the planes XZ, YZ, and on OZ. If in the 
