491 
degree, moreover birational, so that it contains six double rays. One 
of these is the line PS, in which a ray w and a line ¢ coincide. 
The other five double rays meet the same rays of (a), (6), (c) and 
(d) as the corresponding line ¢ and are therefore each conjugated 
as a line / to rays t of >. Hence the order of = is fwe. 
The number of lines of © in a plane W is found by conjugating 
to a ray w of JW the line ¢, which passes through S and meets the 
same rays a and 0, and to ¢ the line w’, which meets the same 
lines c and d as ¢. The correspondence (2v,w’) in W is again 
birational and of degree four, and so has six double rays. Here there 
is no line 2 coinciding with a line ¢; the class of = is therefore sav. 
Hence to a sheaf is conjugated a congruence (5,6). 
To the four lines SA* ete. and to the transversals through S which 
meet the six pairs (AB, yd) correspond plane pencils of lines. Hence 
each congruence = contains ten singular pencils of lines. 
In the same way it is proved that to a plane field of rays V 
corresponds a congruence ® (6,5). 
If S, or V, contains one or more principal rays, this reduces the 
order of ©, or ®, which reduction is easy to calculate in each case. 
6. Two complexes |t} (vide $7 of Prof. pu Vrins’ communication), 
which are conjugated to special linear complexes of lines ¢, with 
axes / and m, have a congruence C (49,49) in common, which we 
are going to investigate. 
In the first place C contains the congruence A conjugated to the 
bilinear congruence Z which has 7 and m for its directrices. Now 
L (4,1) has with a congruence 2 (6,5) eleven rays in common just as 
with a (5,6), from which it follows that A is a congruence (11, 11). 
Moreover each |} contains as double rays the lines of the sheaves 
A, Bb, C, D, those of the planes a, 3, 7, d, and the lines of the con- 
gruence (3,3), constituted by the singular reguli. For, in fact, the 
line / meets two generators of every singular regulus, each generator 
corresponding to the entire regulus. In the intersection of the two 
complexes corresponding to / and m, each of the nine above-mentioned 
complexes is to be reckoned for four. Together they therefore count 
for a congruence (28, 28). 
Furthermore each |/{ contains as single rays the lines of the 
four sheaves A“, etc.. those of the four planes «*‚ ete. and the lines 
of the six bilinear congruences (AL, yd), ete. (§ 4). These form 
together a congruence (10, 10). 
By the foregoing investigation the congruence (49,49) is com- 
pletely accounted for and the completeness of the discovered system 
of singular rays is controlled. 
