492 
7. By virtue of the HarpneN theorem the congruences >, which 
are conjugated to two sheaves P and Q have 61 rays in common. 
To these belongs in the first place the line ¢’ corresponding to the 
linet — 20) 
Each congruence has among its generators the principal rays AB 
etc. (6 in number), ag etc. (6), AA* ete. (4). For, in fact, to each 
of the twelve former lines corresponds a bilinear congruence, to the 
latter four a sheaf and all these have one ray which passes through 
an arbitrary point. This accounts for 16 of the common rays. 
The remaining 44 are found as follows. According to § 1 there 
exist reguli (bed) which pass through A and also through P and 
Q. As follows from $ 5 of Prof. pw Vries’ communication a regulus 
(bed) through A is conjugated to a ray passing through A. Hence 
the two congruences = have in common five rays passing through 
A and as many passing through B, C and D resp. 
Furthermore there are six surfaces (bcd) which are tangent to @ 
and pass through P and Q. A surface (bed) which is tangent toa, 
is conjugated to a line lying in e. Hence the two congruences 2 
bave in common six rays lying in « and six in B,y and d each. 
So in this way we find indeed 4 «5+ 4 « 6= 44 additional 
common rays. 
The investigation of the intersection of two congruences ® is 
quite analogous. 
Slightly different is the case of the common rays of the congruences 
= and ® which are conjugated to a sheaf P and a plane of rays 
V, which by virtue of the Hanpsen theorem have 60 rays in 
common. 
To these common rays belong the principal rays AB (6), «B ete. 
(6), 12 lines in all. 
There are six surfaces (be dl) passing through A and Pand simultane- 
ously tangent to WV. This furnishes six lines through A, common to 
= and ®. The same holds for B, C and D, so we have 24 lines 
in all. 
The six surfaces (bed), which are tangent to a, pass through P 
and are also tangent to V, furnish six common rays lying in a. 
Just as many we find in the planes 8, y and d, together 24. 
By the foregoing enumeration the 60 common rays are indeed 
accounted for. 
This § constitutes a proof of the completeness of the system of 
principal rays. 
