1118 
[J*, A*] =k + 4, because J has with A & gu points in common, 
We find therefore 
Za*#—2—g94+ 6d=k4-pw. woe ew 2. (2) 
From (1) and (2) it ensues at once 
at —d(n— 4)+4+ 1. 
§ 4. The two particular suppositions we have made are super- 
fluous. Let A* not belong to a linear system of which all the curves 
in the points B, have the multiplicity A. For such an isolated curve 
A* a positive or negative integer g is always to be defined, which 
is called the virtual degree) of A*. An infinite number of linear 
systems may be construed, in such a way that A* is a part of 
curves belonging to it. Let |/*| be such a system and let A be a 
curve that completes A* into an £*. It may be seen to that there - 
are an indefinite number of such curves R*. They form then a linear 
system |/*|, the restsystem of A* with regard to |L*|. Let g, be 
the degree of |Z*| in other words the number of variable inter- 
sections of two Z*, and g, the degree of Rt. If now the A* also 
formed a linear system |A*|, then we should of course have, g being 
the degree of it: g, =g +9, + 2, where 7 represents | A*, R*]. 
For. gy sis [A* BART in whiche 4,* ‘and: \*\-are 
arbitrary curves of |A*| and |&*|. 
If A* is isolated then its virtual degree, by definition, is the 
number g determined by the equation g, = g + 9, + 22. 
This virtual degree g is independent of the choice of | L*|. We 
may further prove that we may calculate with it as if g were “the 
number of intersections of A* with itself”, independent of its posi- 
tive or negative sign. The formula (1) holds good if A* is isolated: 
in that case g represents its virtual degree. 
The same holds true of the formula (2), not only if A* is isolated, 
but also if no canonical curves exist. In all cases 22*—2—g is 
called the immersion constant of A* and is |.J*, A*] — 3|C*, A*]?). 
§ 5. If 2 consists of two different developable surfaces 2, and 
&2,, A* consists of two different curves A,* and 4,*, which both 
have the same deficiency 7 as A. A,* and A,* intersect each other in 
the & images, of the pinch-points on A. Without nearer determina- 
tions it cannot be said that the formula 2* = d(n—4)+ 1 holds 
good, because A* is degenerated. But it may be supposed that the 
1) F. Enriquss, Introduzione, p. 28. 
2) F. Sever, Il genere aritmetico ed il genere lineare, (Atti della R. Acc. d. 
Se. di Torino, vol. 37, 1901—2). 
