DR. PLUCKER ON A NEW GEOMETRY OE SPACE. 
735 
V/e may represent the point corresponding to the given plane (tf', u\ v\ w') by its 
equation, 
(C v! — Bv' — Dw')t — (Ctf—Av 1 — E w’)u + (B?f — A v! — F w')v + (Dt' — E u'+ F d)w =0, (15) 
which may be written thus, 
A (v'u—u'v) + B( t'v -v't)-\- C (u't — tfu) + D(t'w — w't ) + E (w'u — u'w) -f- F(i/w — w'v) =0.(16) 
19. It is easily seen that both equations (12) and (16) are the most general ones, 
indicating the supposed correspondence between point and plane. Therefore (10) is 
the most general equation of a linear complex. 
20. According to the fundamental relation which characterizes a linear complex, the 
plane corresponding to a given point is determined by means of any two rays passing 
through that point, as the point corresponding to a given plane is determined by any 
two rays confined within that plane. 
Suppose P and P' to be any two points of space, and p and p' the two corresponding 
planes. Let I be the right line joining both points, II the right line along which both 
planes intersect each other. Draw through I any plane intersecting T I in Q, join 
Q to P and P' by two right lines QP, QP'. These right lines, both passing through 
points (P, P') and falling within planes (p, p') which pass through them, are rays of the 
complex. The plane PQP', containing both rays and consequently containing I, corre- 
sponds to the point Q, whence we conclude that planes passing through any points 
Q, Q! of II intersect each other along I. Likewise it may be proved that any plane 
drawn through II intersects I in the corresponding point. We shall call I and II two 
right lines conjugate with regard to the linear complex , or merely conjugate lines. The 
relation between two conjugate lines is a reciprocal one ; each of them may be regarded 
as an axis in space around which a plane turns while the corresponding point describes 
the other ; each also may be regarded as a ray, described by a moving point, the corre- 
sponding plane of which turns around the other. 
Each right line meeting two conjugate right lines is a ray of the complex. 
To each right line of space there is a conjugate one. 
If a point move along a ray of the complex , the corresponding plane — containing each 
ray of the complex which passes through the point, and therefore especially the given 
one — turns around the ray. 
Each ray of the complex may be regarded as two coincident conjugate lines. 
21. We may also connect the preceding results with the general principle of polar 
reciprocity. Indeed the general equation (10), which represents the plane correspond- 
ing to a given point, is not altered if x 1 , y\ z' and x , y, z be replaced by one another. 
Consequently we may say, in introducing the denominations pole and polar plane 
instead of corresponding point and plane, that the polar planes of all points of a given 
plane pass through its pole, and conversely, that the poles of all planes passing through 
a given point fall within the polar plane of that point. In our particular case a plane, 
5 i 2 
