748 
DE. PLUCKEE ON A NEW GEOMETEY OF SPACE. 
If there exist a point (x, y, z) where all rays of the complex meet, this point will be 
determined by means of the following three equations, 
A— Fy— Es =0,j 
B + F^-D2=0,i (56) 
C+Eff+Dy=0.J 
These three equations can subsist simultaneously only in the case where (55) is satisfied. 
If this condition be satisfied, the locus of points, where all rays of the complex meet, 
is a right line, the projections of which are represented by the last equations (56). 
46. Such rays as belong to both linear complexes, 
Q,z=zAr + Bs + C 4-D<r +Eg> +F(s§ — 
0'=A !r + B's + C' + DV + E 'g -f F'(s ? -r<r)= 0, { ‘ 
constitute a linear congruency of rays represented by the system of the two equations. In 
order to determine the congruency each of the two complexes, 
0 = 0 , O '=0 
may be replaced by any other represented by 
O+^Q'=0, (58) 
where arbitrary values are given to the coefficient (m. 
In each of the two complexes by means of which the congruency is determined, there 
is a plane corresponding to each point of space which contains all rays starting from that 
point. Both planes corresponding to the same point intersect each other along a single 
ray, belonging to both complexes, i. e. to the congruency. With regard to the congruency 
one ray corresponds to a given point of space. The planes corresponding to the same 
point, in all complexes, represented by (58) meet along a fixed line, the corresponding 
ray of the congruency. 
Conversely, there is in each of the complexes (58) a point corresponding to a given 
plane in which all rays confined within the plane meet. By means of two such com- 
plexes we get, within the given plane, two points ; the right line joining the two points 
is the only ray of the plane common to both complexes, and therefore belonging to the 
congruency. We call it the ray of the congruency corresponding to the given plane. 
To each point, as well as to each plane, corresponds only one ray. There are not any 
two rays of the congruency intersecting one onother, or, in other terms, confined within 
the same plane. 
47. Suppose that AB is any given right line, and A'B', A"B" its two conjugate with 
regard to the complexes O, O'. Let C be any point of AB. Each ray starting from C, 
if confined within the plane A'B'C belongs to O, if confined within A"B"C to O'. There- 
fore the intersection of the two planes A'B'C, A"B"C, i. e. the right line starting from C 
and meeting both conjugate, is the ray of the congruency which corresponds to the 
point C. If C move along AB, all rays of the congruency obtained in that way are the 
