1119 
curve A,*-+ A,* belongs to a continuous system and in that case 
the curves of that system are of the deficiency 
amt—anatastk—lae2na2stk—l, 
And also if 4,* + A,* does not belong to such a system 2 a + 
i:—1 is called the virtual deficiency of this degenerate curve '). Let u, 
be the class of 2, and uw, the one of 2,. A = intersects A in 
its d(m—1) nodes. They are represented in d(n—1) pairs on /’* and 
of each pair one point lies on A,*, the other on A,*. Hence 
LE*, AT =[E* A,*] = d(n—1). 
And as | =*| = |A,* + A,* + J*|, we have 
d(n—1) = [A,* + At HIS AE =[A,* + A,* HJA AM 
Consequently 
d(nm—1) =a, tk+k + u,=9,+h +h 4+5p,. . . (1) 
where g, and g, are the virtual degrees of A,* and 4,*. 
The immersion constant 2%—2—y of A,* is equal to 
[J*, 4,*] ae [Cr A Ped 
hence 
ktwu,—3d=2a—2—4, (2) 
and glt SR EE gi) Nie PDS NE 
From (2!) it ensues 
ag ek SES lg tds ht tH 3a I. 
Consequently with regard to (1) 
2%+k—l=—d(n—4)4 1," 
The formnla «* =d(n—4)+1 holds consequently good if 2 
degenerates, provided the virtual deficiency is taken for a*. 
So we have this general proposition : 
The order of an algebraic surface that has no other singularity 
but a nodal curve A of order d, along which the pairs of tangent 
planes form a developable surface 2 of deficiency a*‚ is 
n*—|] 
n—=d + ET, 
? 
1) Cf. eg. E. Prcarp “Théorie des fonct alg. de 2 var.” vol. 2, page 106. 
