( 256 ) 
But the following must be pointed out particularly: the surface 
of the third class out of n°. 4 is connected in a simple way with 
the enveloped space. In the cone by which this surface is projected 
from the point P taken there, we have namely before us the envelope 
of all the tangent planes of the cubic space, passing through this 
point P. This will be clear if we resume in the following form 
the dualistically opposite results forming an extension of the theorem 
of SEGRE mentioned in n°. 1: 
If we take quite arbitrarily in S, four planes as, 
O14. A5 Cio, and if we determine the planes eos, G4. G51 
026, forming with the former one a double four 
Q@i3 » G14 » Ayn » A76 
G93 sv Bon 9 Gop v ag 
— where two planes have a point or a line in common 
according to their symbols having a common index 
or not —, then the four points of intersection of the 
opposite elements of the double four — placed here 
under each other — lie in a same plane «jo. 
If we add this plane aj, to the assumed planes, we 
obtain a quintuple of “conjugate planes” with the re- 
markable property, that each of those planes plays 
the same part in reference to the double four, of which 
the four remaining planes form one of the two qua- 
druples, as aj in reference to the above mentioned 
double four. 
If we complete all quadruples to be formed out of 
this quintuple to double fours, we find fifteen planes 
in all, which can be characterised by symbols ars in 
such a way, that two planes have a line or point in 
common according to their symbols having a common 
index or not. We then find that ordering of those sym- 
bols in form of determinants — as the corresponding 
a. in n®%. 4 — gives six rows or six columns of conju- 
gate quintuples (g;),7=1,2,..6. Each line intersecting 
four planes of aconjugate quintuplealso cuts the fifth. 
Each triplet of planes (a, @g4, Qe), Cutting each 
other two by two in a line, lie in a same space indi- 
cated by Sis, 34,56; In this space they pass through a 
same point Pis,s4,56. There are fifteen of such spaces 
and points. 
