Chap. I LINEAR SYSTEMS OF CURVES 255 
the residual series would be a complete special grf~i 3 . If we add 
an arbitrary point P, we have a complete special grf“i 2 by 8] 
and 9], and that would mean that all special ad joints went 
through this arbitrary point. 
Theorem 14] The canonical series has no fixed points. 
There are cases where the special ad joints have fixed points 
not on the base curve. If the latter be of genus 2, the special 
adjoints form a pencil, and when n > 5 this will have centres 
other than the singular points of the base curve. 
Theorem 15] The order and dimension of a series are invariant 
for birational transformation. 
Theorem 16] The canonical series is birationally transformed 
into the canonical series of the transformed curve. 
Theorem 17] A special series is birationally transformed into 
a complete series. 
§ 2. Sums and differences of series 
Suppose that we have a complete series of order N-{-N' and 
dimension s, g s N+ N>, which contains a group G of N points. This 
will be a group of some complete series, let us say a g r N , which 
consists in groups cut by all adjoints of a certain order through 
a group G. The g s N+N > shall be cut by all adjoints of a certain 
order through a group P. The adjoints of this latter system, 
which contain G, will cut a complete g r N > whose groups are 
co-residual to G and so residual to all groups of the g r N . The 
complete series will contain every group of g r N and every 
group of g r f' and so be their sum, as defined in Ch. VIII of 
Book I. 
Theorem 18] If a complete series contain in one of its groups 
a group of a series of lower or equal order, it contains the complete 
series containing this group. 
We shall say in this case that the complete series of larger 
order ‘contains’ that of lesser order, and the latter is ‘contained’ 
in the former. The complete series whose order is the difference 
of their orders, and which contains groups obtained by taking 
groups of one series from the corresponding groups of the other, 
is also included in the series of larger order, and is defined as the 
‘difference’ of the two series. Each of the series of lower order 
is said to be ‘residual’ to the other in the series of larger order.