LINEAR SYSTEMS OF CURVES 
253 
Chap. I 
each group. Let the complete series be cut by the totality of 
special adjoints through a group (perhaps empty) G, and let 
an arbitrary line through P meet the base curve again in 
Qi, Qz’--’ Qn-i- There is certainly one adjoint of order n—2 
through the group G and the point P, the totality of adjoints 
of that order through those points will cut a complete series. 
But as such adjoints contain n— 1 collinear points, they consist 
in the line through those points and a special adjoint, and the 
complete series they cut will consist in the original series, 
residual to G, and the fixed point P. 
Reduction Theorem 8] If a fixed point be added to each group 
of a special complete series, the resulting series is complete. 
Suppose, now, that we know about a certain g r N that 
N—r^p— 1. 
We may suppose the series complete, for the inequality will be 
strengthened when we pass from an incomplete series to a com 
plete one. If r — 0 the series is certainly special, for we have 
seen that the dimension of the canonical series is certainly as 
great as p— 1, so that we can always pass a special adjoint 
through p— 1 points. Suppose that we have proved that for 
every series of dimension less than r, if the order less the dimen 
sion be not above p— 1, the series is special. Consider the 
adjoints which cut our series and which pass through an 
arbitrary point, not common to all of them. They will cut a 
which must be special, since {N—l)—{r—l)^p—l by 
hypothesis. It is also complete as it is cut by the totality of 
adjoints through certain fixed points but otherwise not re 
stricted. The special adjoints which cut it must all pass through 
the arbitrary fixed point, for if they did not, we should get 
a complete g r ^ 1 by adding the point to the gjT-u whereas there 
is a complete g r N . Since this point is arbitrary, any group in the 
original series can be cut by a special adjoint. 
Special Series Theorem 9] If the difference between the order 
and the dimension of a series be less than the genus of the curve, 
the series is special. 
Theorem 10] If a complete series be not special, the difference 
between the order and the dimension is equal to the genus of the 
N—r — p. (3) 
curve