DR. PLUCKER ON A NEW GEOMETRY OE SPACE. 
751 
(D'E-E'D> 2 -[(B'E-E'B)— (A'D-D'A)>- (A'B-B'A)=0, . 
(B'D - D'I% 2 + [(B'E - E'B) + (AD - D'A )~\ccy + ( A'E - E'A)^= 0, 
(AD — D' A)y + (A'E — E' A)ar + (D'E — ED )yz =0, 
(BD-D'B)y-(B'E-E'B)#-(D'E-ED).r2=0, 
(A'B — B'A)y + (A'E — E'A)#2 — (B'E — E'B)y2= 0, 
(A'B - B'A> - (AD - D'A)xz + (BD - DB>2= 0 * 
(65) 
( 66 ) 
(67) 
( 68 ) 
(69) 
(70) 
53. According to the first two equations (65), (66), the locus in question is a system 
of two right lines both intersecting OZ. These lines are confined within two planes 
parallel to XY and determined by (65) ; their direction within these planes is given by 
(66). We shall call them the “ directrices” and the characteristic section parallel to 
both and equidistant from them, the central plane of the linear congruency. Both 
“directrices” intersect at right angles the axis of the congruency, as the axes of all 
complexes do. 
54. We may distinguish two general classes of linear congruencies ; either both direc- 
trices are real or both imaginary. In a particular case the two directrices are con- 
gruent. Finally, one of the two directrices may pass at an infinite distance. 
55. If the directrices are real, and the plane XY be conducted through one of them, 
the following condition, A'B— B'A— 0 (71) 
is derived from (65). In order to determine within XY the direction of that directrix, 
we get from (67), by putting 2=0, 
(A'D-D'A)y+(A'E-E'A>=0 (72^ 
There is among the infinite number of complexes containing the congruency, which 
are represented by 
12-f-jO«Q'=0, 
one of a particular description. It is obtained if, starting from (62), we put 
whence 
A B . 
(AD - D'A)* + (A'E - E'A)g = 0. 
(73) 
All rays of that complex, and therefore all rays of the congruency, meet within XY a 
fixed right line, represented by (72), on replacing g and a by x and y. This line there- 
fore is the axis of that complex, and one of the two directrices of the congruency. In 
the same way it may be proved that likewise all rays of the congruency meet the other 
directrix. Hence 
All rays of a congruency meet its two directrices. 
* We may observe that any equation which, like those above, is homogeneous with regard to (A'B— BA), 
A'C — C'A) . . . will not be altered if the complexes 12 and 12' are replaced by any of the complexes (12+jul2'). 
