750 
DR. PLtTCKER ON A NEW GEOMETRY OE SPACE. 
Accordingly the equation of the section becomes 
(E'F — F'E)# — (D'F —E'B)y (D'E — E!B)z = 0, 
and the double equation of the right line perpendicular to it, 
x —y z 
E'F — F'E D'F — F'D D'E— E'D * ' ’ * 
(59) 
(60) 
51. By giving to OZ the characteristic direction, the two complexes (57) will be 
represented by linear equations of the form 
0=Ar +C 4-D<r +Eg> =0,1 
Q'=AV H-B's-f O' +D'<r+ E'g = Oj 
(61) 
the origin and the direction of OX and OY, perpendicular to OZ, remaining arbitrary. 
Again, OZ may be moved parallel to itself, and accordingly o and a replaced by (g +# 0 ) 
and ( G-\-y ° ), x° and y° being the coordinates of the new origin. If especially 
C+D/+Etf°=0, 
C'+Dy+EV=0, 
whence 
B C'D-D'C 
* ~ “D'E-E'D’ 
C'E-E'C . 
y — D'E-E'D ’ 
by the mere disappearance of C and C' the equations of the two complexes become 
O =Ar -j-Bs -j-Eg =0, j 
Q'EEAV+B's+D'<r+E'e = 0.J 
OZ in its new position is a completely determined right line, which may be called 
the axis of the congruency. It is easily seen that it intersects at right angles the two axes 
of rotation of the complexes Q and O', and consequently the axes of all complexes 
represented by (58). 
52. The planes corresponding in the two complexes (62) to a given point (x’, y\ z') 
are represented by 
(A -E z' >r+(B —~Dz' )y+(EF -{-By' )z=Ax' +B y', | 
(A'-EV>+(B'-DV)y+(EV+Dy>=AV+B^'./ ‘ 
In order to express that both corresponding planes are the same, we obtain the fol- 
lowing relations, 
(A — Es') : (B —Hz') : (Ex' +Ey') : {Ax' +By')=l Q 
(A'-EV) : (B'-DV) : (EIx'+E'y 1 ) : (AV+B'y). J } 
Since both planes pass through the given point, any two equations, hence derived, are 
sufficient in order to determine the locus of points having, in both complexes, the same 
corresponding plane. From any two of the following six equations where the accents 
are omitted, the remaining four may be derived ; 
