1139 
Through each of these points pass five curves c°‚ so that this point 
belongs to five points P of the line /. The lines connecting these 
points with P bear each a coincidence lying in Q, so that every 
time a point of c*® coincides in Q with an associated point of 2. 
Each of these 9 points of Q being a fivefold point of the curve A, 
45 intersections of 24 and V are found. 
The total number of intersections of 4 and V amounts therefore 
to 6+ 21 + 45 — 72; this therefore is the order of 2. 
A second plane V" intersects the curve 4 in 72 points; there are 
consequently 72 groups of S*, of which two points lie in two given 
planes, whilst a third lies on a given straight line. 
§ 10. As appears from the preceding there are oo? groups of S‘, 
of which two points lie in two given planes Wand V'. The remain- 
ing points of these groups form a surface of order 72, for a line / 
contains 72 of these points. 
Among these groups there are o' that have a coincidence lying 
outside the planes V and V'. The locus of these coincidences is a 
curve o, the order of which we shall determine. 
With a view to this we try to find the number of intersections 
of the curve @ with the plane V. 
The plane V' intersects the plane V along a line /. The latter 
contains 21 points P, to which belong a coincidence Q lying in the 
plane V and a second point of the plane I’, the 21 points are the 
intersections of the straight line / with the surface ®, which belongs 
to the plane WV. The 21 associated points Q are evidently intersec- 
tions of the plane | with the curve g. 
The plane JV intersects the curve #° in 9 points Q, There are 
w' groups of S*, of which three points coincide in Q; as appeared 
in $ 6, the locus of the remaining points of these groups is a 
biquadratie twisted curve. It intersects the plane V' in four points. 
There are consequently four groups of which a point lies in WV, 
while the three other points have coincided in the intersection of 
the support with the plane WV. This may evidently be considered 
in such a way that a point of the plane V has coincided with a 
coincidence associated to it; each of these groups produces therefore 
an intersection of the plane V with the curve d. The number of 
these groups amounts evidently to 9 > 4—= 36. 
The order of d therefore amounts to 21 + 36 — 57. 
A third plane VV” intersects the curve 6 in 57 points. There are 
consequently 57 groups of S', which have in a given plane V" a 
comcidence, while the two other points lie in two given planes V and V". 
73 
Proceedings Royal Acad. Amsterdam. Vol. XIX. 
