1140 
§ 11. The surface of order 72 formed by the remaining points 
of the groups of which two points lie in two given planes V and 
V', is intersected by a third plane V" along a curve c?*. There 
are consequently o' groups of S* of which three points lie in 
three given planes. The fourth points of these groups form a curve 
u, of which we shall determine the order. To this purpose we try 
to find the intersections of the curve u with the plane V. 
The plane JV intersects the planes V' and WV" along two lines 
/' and /". The surface 4°’, which belongs to the line /’, is intersected 
by the line 7" in 26 points. Two of these points lie on the line /', 
which is a nodal line of A**; the 24 remaining ones determined 24 
groups of JS‘, of which two points are respectively lying on the 
two lines /' and 7". - 
The supports of these groups lie in the plane V,and the remaining 
two points of each of these groups are intersections of the plane 
V with the curve u. In this manner 48 intersections are found. 
There are 57 groups of S* that have a coincidence in |, while 
the two other points of those groups lie in the planes V' and WV. 
In each of these coincidences a point of V has coincided with the 
associated point of w; in this way 57 coincidences of V and u are 
found. 
The plane JV intersects the curve o° in 9 points. Each of these 
points Q bears oo? coincidences of S*‘; the remaining points of these 
groups lie on the polar surface 7° of the point Q. This surface 
intersects the plane V' along acurve 7’; among the groups mentioned 
there are consequently o', of which one point lies in the plane V; 
the remaining points of these groups form a curve 7. This curve 
y, intersects the plane J”' in the 9 points, in which WV’ intersects 
the curve gp’ for, in each of these intersections the corresponding 
group has a coincidence, so that there a point of J” coincides with 
the corresponding point of 1. The curve 7 is therefore of order 9. 
The plane VV" intersects the curve 7’ in 9 points. With each of 
the 9 intersections of V and v° 9 groups are consequently found, 
which have a coincidence in the intersection mentioned, while the 
two other points lie in the planes V' and VV". It is easy to see 
that these coincidences are in their turn intersections of the plane 
V with the curve u; in this way 81 intersections are found. 
The total number of intersections of V and u, amounts therefore 
to 48 + 57 + 81 =186. This, therefore is the order of wu. 
A plane VV" intersects the curve u in 186 points. There are 
consequently 186 groups of S‘, of which four points lie in four 
gwen planes. 
