( 257 ) 
Each sextuple of points as ‘(co G13) @3, Cass Cage 56), 
built up of two triplets (eo, 3) @3) and (eas Caos &55)) With 
the property that each plane of one triplet cuts each 
plane of the other in a line, pass through a same point 
Piz, 456. There are ten such points. 
A threedimensional space S; taken arbitrarily cuts 
each of the six conjugate quintuples (g;) in five lines, 
which admit of two common transversals (bi, ¢); these 
six pairs of lines (b,c) are opposite elements ofa 
double six of a surface £8 of order three of which the 
27 right lines consist of these twelve lines and the 
fifteen lines of intersection of S with the planes ak, 
The locus of the lines intersecting four planes 
belonging to a same conjugate quintuple — and so 
also the fifth — is always the same curved space S34 
of order three and class four through the fifteen 
planes az, whichever of the six quintuples (g,) are 
taken; so this space S% contains six different twofold 
infinite systems of right lines. It has the ten points 
Pis, 456 as double points, the quadratic conic spaces of 
contact of which contain the sextuples of planes 
passing through those points; it is cut into three planes 
by each of the fifteen spaces Sia, 34,56. Its section 
with the above introduced arbitrary space Sz must 
contain the double six of the pairs of lines (b;, ¢) as 
well as the fifteen lines of intersection of 8S; with the 
planes az, and so it must coincide with the surface F° 
found there, of which the points lying outside these 27 
lines are points of intersection of S; with lines of the 
locus S34 not situated in Ss. 
7. If we apply to the equations 
Sn iu ate NH 
