( 262 ) 
Out of the fifteen triplets of equations 
Pit Pe=9, pp + pa = 0, ps + ps =O 
it directly ensues that the curved space of class three enveloped by 
the pencils of planes with one of the lines of the configuration as 
axis must have the equation 
Pi’ + Pa° + Ps + Pa? + Ps? + po = 9 
the first member disappearing for each of those triplets. 
So the six points p;=0 are for this curved space of class three 
what the pentaeder of SYLVESTER is for the surface of order three. 
9. In asecond treatise published in 1891 and entitled: ,Ricerche 
di geometria delle rette nello spazio à quattro dimen- 
sioni” (Atti del. R. Istituto Veneto, series 7, vol. 2) CASTELNUOVO 
has represented the curved space 
=P Ha + ag + 248 + 258 + 20° = 
on our space Ss; in such a way, that the spacial sections of g? = 
correspond with quadratic surfaces passing through five fixed points. 
In that case the fifteen planes correspond to the five vertices 
and to the ten faces of a complete quintangle in Ss, whilst the 
ten double points of g’=0 correspond to the ten edges of this 
quintangle. Instead of continuing these researches we put the 
question in how far the configuration of fifteen lines is unique in 
its kind. 
Of course it is not difficult to point out in the poly-dimensional 
spaces configurations having characteristics in common with the 
configuration of SEGRE. So we find one in each group of n + 2 
points taken arbitrarily in S, when n-+ 2 is not a prime number. 
Let us take as an example nine arbitrary points 1, 2, 3,.., 9 
in S, and let us represent the point of intersection of the plane 
(1, 2,3) with the space §; through the six remaining points by 
the symbol Pj23; then each three points Pigs: Py, Pys9, whose 
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