tLET 
by the images of the sections of / with arbitrary surfaces of 
order n—1. Let us now for a moment suppose that A* belongs to 
a linear system on /* of which all curves pass as often though 
the different points Bj, as A*, and let A,* be a curve of that linear 
system. In that case A,* -+ J* is also a curve of |Z*|. Let us for 
convenience’ sake represent the number of intersections of two curves 
outside the points B, by placing the letters we have chosen for those 
curves, between brackets, we have 
[=*, A*] =[A,, A*] + [J*, A+). 
[A,, 4*] is called the “degree” g of the linear system to which 
A* belongs.') J rests in kJ u points on A, consequently 
Us, A*] =k +a 
= has d(n—1) nodes on A, therefore [>*, A*| = 2d (n—41). 
Hence, 
ACM CEES En NA TENEN 
We obtain a second relation if for a moment we make a parti- 
cular supposition: let there exist surfaces g”—* passing through the 
nodal curve A, consequently adjuncts of order n—4 of £. They 
intersect /#’ apart from A in so-called canonical curves K, which 
have the property of being represented on /’* as canonical curves 
K*, consequently, as sections of #* with adjuncts of order n*—4. 
Two properties of the canonical curves A we have to apply here. 
An adjunct g"—* forms with 3 planes an adjunct g"—! of order 
n—1. To gp” belong also the 1st polar surfaces of arbitrary points 
of space. So a K forms together with 3 plane sections C a curve 
of |J|. Consequently a A* forms together with 3 curves C* a J*, 
so that 
[K*, A*] + 3 [C* A*] = [J *, A]. 
The second property we want, we find by observing that an 
‘adjunct grt forms with 1 plane an adjunct ¢”—%. A gr intersects 
the plane of a C in a curve g”~—*, which passes through the d nodes 
of C so that outside it, it has moreover 2p—2 points in common 
with C, where p is the deficiency of C. Hence the canonical curves A 
intersect Cin 2»—2—n points, where n is the degree of the linear 
system of the C. But this holds good for any linear system of 
curves. *). Let us apply this to the system to which A* on F* 
belongs, we have then | K*, A*] = 2a*—2—g. 
Further is [C*, A*]— 2d, because C has d nodes on A, and 
1!) Cf. eg. F. Enriques, Introduzione alle Geometria sopra le superficie alge- 
briche. (Memorie di mat. e di fis. d. Soc. It. d. Se., Serie 3, volume 10, p. 14) 
2) F, Enriques, Introduzione, p. 64. 
71 
Proceedings Royal Acad. Amsterdam. Vol. XIX. 
