738 
DR. PLUCKER ON A NEW GEOMETRY OF SPACE. 
Accordingly the coordinates of these points are 
C 
B 
x— 0, 
y= 
Z — — 
C. . 
A 
y=o> 
x= 
Z ~E ==Z «’ 
B 
A 
z = 0, 
x= 
P = 5 
y=Y=yv, 
whence may be derived the following relation, 
x v y t 2 u _ _ j 
In putting C= — 1, the right line conjugate to OZ, if regarded as an axis, may be 
determined by its four coordinates [5], 
j)=A, c[— B, bt=D, ^=E. 
These coordinates therefore are four of the constants of the complex 
Ar-f-Bs+D<r-t-Eg+F(sg— s<r)=l. 
MN conjugate to OZ remains the same whatever may be the value of F. If by putting 
F equal to zero the last equation becomes a linear one, the complex is completely deter- 
mined by MN conjugate to OZ. 
26. The ratio of the three constants upon which the characteristic direction of the 
linear complex (1) depends, D E F 
remains the same if the origin be changed or the complex moved parallel to itself. But 
if by turning the complex the characteristic direction simultaneously move, that ratio is 
altered. One of the three constants F, E, D becomes zero if the characteristic direction 
be confined within XY, XZ, YZ ; two of them disappear, F and E, F and D ; E and D 
if that direction fall within OX, OY, OZ. Here the general equation becomes 
Ar+Bs + C+Dff =0, 1 
Ar+Bs+C+E ? =0, l (22) 
Ar+Bs+C+F(s§— r<r) = 0. j 
27. The ratio of the three constants 
A : B : C 
varies it the complex be moved parallel to itself. If the plane corresponding to O pass 
through OZ, OY, OX, one of the three constants C, B, A becomes zero ; if this plane 
be congruent with XY, XZ, YZ, i. e. if O be the point corresponding to XY, XZ, YZ, 
two constants A and B, A and C, B and C disappear, and the general equation of the 
