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laid out of P at F. J is the section of / with the first polar surface 
of P, apart from A. From this it ensues that the sections + of F 
with arbitrary surfaces of order n—1 represent themselves as a 
system of curves =*, individuated by the compound curve A*+./*, 
As the fare of order », the order of A*-+J* is equal to (n— 1). 
J* (as jacobian of a net of curves of order v) is of order 3(~—1). 
Hence A* is of order 
(n—1) v—3 (rd) = vo(n—4) 3. 
The curves C* may have base points. If B, is an h-fold base 
point, in such a way that all C* pass A-times through Bj, the 
Jacobian J*, as is known, passes 34—1 times through 4,. The 
section of # with an arbitrary surface of order n—1 is represented 
on F'* as a curve 2, which is represented by a homogeneous 
equation of order n—1 in the /;, so that such a curve passes (n—1)h 
times through By. Hence A* passes (2—1)h—(8h—1) = h(n—4) +1 
times through Bj. The deficiency a* is easy to calculate now. We 
have viz. 
«== 4 fv (n—4) + 23 f (n—4) + 1} — J $1 (n—4) th (n—4) +1), 
in pa the summation extends over the various base points Bj. 
If we consider that the deficiency of a plane section C of F is 
equal to that of its image C* on #'*, in other words that we have 
L (pl) (vw —2) — $ Bh (h—1) = $ (n—1) (n—2) —d, 
we find 
a* — d(n—4) + 1. 
3. The above reasoning can be of no service if /’ is not rational, 
so that #’* is not a plane. Let /’” be a surface of order n, rational 
or not, with a double curve A of order d and without oi other 
singularity. Let u be the class of the developable surface 2 formed 
by the pairs of tangent planes along A and let & be the number of 
points of A, where the two tangentplanes coincide (pinch-points). 
Suppose that F has been transformed into another surface /’ by 
a birational transformation into another surface /’*, in such a way 
that A passes into one single curve A*. The plane sections C' of # 
are represented by curves C*, which form a linear system on f*. 
These C* may have base points B, so that they all pass / times 
through Zj. The sections of / with the oo? planes passing through 
a point P are represented by the curves of a net (C*). The curve 
of contact J of the cone of contact laid at / out of P is trans- 
formed into the Jacobian J* of (C*), from which it follows again 
that A*-+.J* is a curve belonging to the linear system |2*| formed 
