798 
of conies through 8—D_ points cuts c, in an algebraical sequence 
of points get (order = D—2, dimension —= D—8). Whenever a 
group of these is contained in a group of y, a degeneration appears 
in the net. The number of times this happens is found from the 
formula 
where n indicates the order, r the dimension of g, im the order, 
v the index of y, d the number of double points (here d = 2D) ’). 
In this case we find accordingly z= D—2, in other words 
through each base point pass D—2 straight lines, parts of degene- 
rations. In all (9—D) (D—2). 
c. A straight line through none of the base points is completed 
by a conic through 9— D base points. The system of conics defined 
by these points cuts c, in a 95 3. By the aid of the same formulae 
we have z= } (D—2)(D—3), which represents the number of 
degenerations where the straight component does not pass through 
any of the base points. 
The total number of. degenerations is accordingly *) 
} (9—D) (8—D) + (9—D) (D—2) + 4 (D=2) D=3) 228 
2. From the preceding follows that in a net of cubics with 6 
base points A,....,A, through each base point there passes one 
straight line which is completed to a degeneration by a conic through 
the other 5. It is known that these 6 straight lines pass through 
one point P when 4,...., 4, lie on a conic c,. Besides (he degene- 
rations PA;-+ c, are in this case contained in the same pencil of 
the net. All the nets chosen from the complex (threefold infinite 
linear system) of cubics defined by A,...., A, have this property, 
hence also the net with the base points P,A,,...., 4,. The existence 
of the fundamental curve c, causes this property. 
We shall now investigate whether this singularity can also appear 
in nets where there is no fundamental conic. 
Let A,,....4, be the base points of a complex S, and let us 
curva algebrica’’, Atti del Reale Instituto Veneto, t. 67% p. 1323, (1908). 
2) ©. Segre, „Introduzione alla geometria sopra un ente algebrico semplice- 
mente infinito”; Annali di Matematica, Ser. Il, t. XXII, p. 41. 
3) That the number of degenerations amounts to 21, independently of the 
number of single base points, follows also from the considerations in the paper 
“On Nets of Algebraic Plane Curves” (JAN pe Vries, these Proceedings VII (2), p. 716. 
