1637 
On any surface ® of S, an (r—1)-fold infinite system of twisted 
curves is produced by the intersection of the remaining ones, of 
which system of twisted curves we will suppose that any of them 
is of the genus one. 
Jf we take r>>2 a net of elliptical twisted curves arises at 
least on each surface d, so that any surface of JS, is rational, i.e. 
may be represented point for point in a plane‘); in such a repre- 
sentation the system of twisted curves will have to correspond to 
the likewise (r—1)-fold infinite system of plane elliptical curves 
(r > 2). Suchlike linear systems of plane elliptical curves may always 
be reduced by means of birational transformation to a system of 
cubies with u single base-points (O< <7) or to a system of quartics 
with two two-fold base-points. *) 
If m is the number of free intersections of two curves (at the 
same time the number of points in which three surfaces of S, meet 
moreover apart from the base-curves and base-points), the dimension 
of the system of curves in each of the cases proves to be equal to 
n; the dimension of the system of surfaces $S, is therefore r—=n-+1. 
2. Between two spaces + and 2’ a (1, n)-fold transformation 
is achieved, if we establish a collinear correspondence between the 
planes of a space 2’ and the surfaces of a triply infinite linear 
S;, of which the elements intersect, apart from the base- 
curves and base-points, moreover in 7 points. 
If S, is chosen out of an 7-fold infinite system, as has been above 
supposed, any surface ® is to be represented in a plane in such a 
way, that the * intersections with the other surfaces correspond 
to a net of cubics or of quartics. We exclude the latter case and 
assume moreover that all the base-points of the c,-system are different 
for the representation of >. 
system S 
3. For the space transformations are now of importance: the 
surface of Jacost, “7, in the space and the limiting surface 
Dy in the space ' corresponding with it point for point. 
Taking into consideration the above mentioned suppositions we 
shall prove that the limiting surface ®'y must always be a cone. 
lts order may be determined in the following way. 
The curve of intersection of a surface ® of S, with dj corre- 
1) M. Noerger, “Ueber Flächen, welche Schaaren rationaler Curven besitzen’’, 
Math. Ann., 3, p. 161, 1871. 
2) G. B. Guccta, “Generalizzazione di un teorema di Noether”, Rendiconti 
del Circolo Matematico di Palermo, tomo I, fase. 3, 1886. 
105* 
