( 263 ) 
indices complete each other to 1, 2, 3... ,.9, will be situated 
on a right line. It may be, that the easiest way of proving this is 
by the aid of a system of parallel forces in equilibrium applied to 
the points 1, 2, 3, . . , 9. As is known the parallel forces applied 
to the points 1,2,3,..,8 may be chosen in such a way that the 
resultant acts on point 9; if we add to these eight forces a force 
applying in point 9 equal and opposite to this resultant, then such 
a system of forces in equilibrium has been obtained. It is now 
evident that point Pos is the point of application ot the resultant 
of the three forces working at the forces 1, 2, 3 as well as that 
of the resultant of the six remaining ones. For, these points 
of application must coincide, on account of the equilibrium, in a 
point situated in the plane (1, 2, 3) as well as in the space S; through 
the six other points. If we reduce the nine forces to three by 
compounding those operating in 1, 2,3 and those operating in 
4,5,6 and those operating in 7, 8,9, we obtain those parallel forces 
applied in Pias, P455. P7s9 and these three forces can only then be 
in equilibrium when the three points of application lie on a right 
line. So the figure of the nine points S; leads to (9); = 84 points 
1 
Pis, situated three by three on = (9)3. (6.3 = 280 lines, whilst 
reversely ten of those 280 lines pass through each of those 84 points. 
So we can deduce out of 12 arbitrary points in Sj), performing the 
decomposition of 12 into two factors in different ways, 66 points 
Pig, 220 points Piog or 495 points Pi»54 and remark that the points 
Py, are situated six by six in 10395 spaces S,, the points Pj93 four 
by four in 15400 planes and the points Pigs4 three by three in 
13305600 lines, etc. 
Although the configuration of SEGRE is undoubiedly a part of 
the general group indicated here, it is certainly distinguished from 
most of them and probably from all of them by the property that 
it is determined by a quadruple of crossing lines taken arbitrarily 
and these lines fix in a narrower sense a fifth of the fifteen lines, 
which group of conjugate lines then bear together the fifteen points 
of the configuration. Indeed, in S, the system of six points as well 
as that of the four lines is dependent on 24 parameters and so 
these figures agree in number of constants, according to an expres- 
sion of Scuupert. If on this point we examine the configuration 
deduced from the nine points of S;, it is even evident by merely 
consulting the numbers of constants 63 and 12 of the nine points 
and of a line in Sy, that it is impossible to determine the 280 
right lines (123, 456, 789) by some of them crossing each other ; 
18 
Proceedings Royal Acad. Amsterdam. Vol. LV, 
