752 
DE. PLUCKEE ON A NEW GEOMETEY OF SPACE. 
Accordingly, both directrices being real and known, we may immediately draw through 
any given point the only corresponding ray of the congruency. 
56. In that peculiar class of congruencies indicated by the condition 
D'E— E'D = 0, (74) 
one of the two directrices passes at an infinite distance. By putting simultaneously 
A'B-B'A=0, 
we get, in order to represent the only remaining directrix, now confined within XY, the 
same equation as before (72). But among the complexes, 
0 +^= 0 , 
there is, besides the complex (73), the axis of which is the directrix, another complex, 
represented by DO f -D'Q=(AT)-DA)r+(BT>-DE>=0, 
the rays of which are parallel to a given plane. Its equation may be transformed into 
Ar+Bs=0; (75) 
accordingly the equation of the plane becomes 
Aa’+B^=0. 
Hence in this peculiar case 
All rays of the linear congruency meet the only directrix , and are parallel to a given plane. 
57. From the last considerations we conclude that among the complexes intersecting 
each other along a linear congruency, and represented by 
O+^O'=0, (76) 
there are in the general case two, of a peculiar description, all the rays of which meet 
their axes. These axes, the directrices of the congruency, are two conjugate right lines 
with regard to each of the complexes (76). 
Generally there is only one ray of the congruency passing through a given point, as 
there is only one ray confined within a given plane. But each of the two directrices 
may be considered as the locus of points, from which start an infinite number of rays, 
constituting a plane which passes through the other directrix. It may be likewise 
regarded as enveloped by planes, confining each an infinite number of rays, which con- 
verge towards a point of the other directrix. 
58. We may represent any two complexes O, O' in any position whatever by equa- 
tions depending only upon the position of their axes and their constants. Let A be 
the shortest distance of the two axes from each other, and S- the angle between their 
directions. 
Suppose that OZ intersects at right angles the axes of both complexes. Let OX be 
the axis of the first complex O, k its constant, OX perpendicular to XZ. The equa- 
tion of the complex will be 7 
1 <7= AT. 
