( 260 ) 
For we recognize in the ten barycentres 12, 13, .. , 45 of two — 
masses and the five barycentres i’, 2',.., 5’ of four of the five 
masses the fifteen points, in the thirty planes (12, 34, 35, 4’, 5’) 
and the fifteen planes (1', 24, 35, 23, 45) the forty-five planes, in 
the ten spaces (12, 34, 35, 45, 3’, 4, 5!) and the five limiting spaces 
of the five-cell the fifteen spaces of the configuration. And if we 
assume in the points 1, 2,3,4,5 entirely arbitrary different masses 
instead of equal ones, then the special case represented in fig. 4 
passes into the general one. If we then also call the barycentre 
of those unequal masses the point 6, we arrive at the following 
simple representation of the configuration : 
If in. space S, six points 1, 2,..,6 are taken in such 
a way that no five of these points lie in a three-dimen- 
sional space, we can find fifteen three-dimensional 
spaces Rs. 2—(3456), Ra(.)—(2456), .. , Rj O=(1234) each one 
of which contains Tour ofthe “six points. “On these 
spaces S&P any three, the indices 7,4 of which complete 
each other to. 1,2... (tovipees through a; sate sans 
furnishing in all fifteen lines. And these same lines 
forming with each other the chief part of a confi- 
guration of SEGRE are also found if each of the lines 
lio, ig» - 5 46 connecting the six points two by two is 
cut by the opposite space R,(12), R,03),.., R68 by which 
operation we obtain fifteen points 12, 13,..,56 charac- 
terized by the property that any three points the 
indices of which complete each other to 1, 2,..,6 are 
lying on a right line. Every five lines containing 
together the fifteen points 12, 13,..,56 form a quintuple 
of conjugate lines. 
8. The simple representation we have now given of SEGRE’s 
configuration is closely related to results published already in 1888 
by Dr. G. CASTELNUOVO in his treatise „Sulle congruenze del 
terzo ordine dello spazio a quattro dimensioni” (Atti 
del R. Istituto Veneto, serie 6, vol. 6). If namely we assume the 
five pointe, 1,2, 3, 4, 5) as” vertices” wp, =DE 
the five-cell of coordinates and the point 6 as point of unity 
Po =(P1 + pa + p3 + pa + ps) = 0, then the pair of equations 
Pi aA Ps == 0e Paste pa 0, 
