508 PROCEEDINGS OF THE AMERICAN ACADEMY. 



e. £. curves of order m that are wound a times around a cone of order 



7)1 — K 



and pass k times through the vertex of this cone. Such curves 



a 



cannot be obtained as the partial or complete intersection of a cone and a 



monoid that have this multiple point as their common vertex. They can 



be obtained as the intersection, partial or complete, of a cone of order 



and a surface of order [i that has the vertex of the cone as a 



a 



(/j, — a)-tuple point ; every line through the (/i — a)-tuple point, and 

 therefore every edge of the cone, meets this surface in a additional 

 points, which are the a points of the curve on this edge. We can avoid 

 doing this by considering these curves as lying on cones of order m or 

 in — 1, as we did in the previous two sections. We shall treat in this 

 section only those curves that are met by an edge of A' m _ K in one point 

 distinct from the vertex. Such curves can be cut out of K m _ K by some 

 monoid of order /a. 



I. Suppose in — k < fx. Then in order to make M^ contain C m with- 

 out breaking up into K m — K and a monoid of order li — m + k, we must 

 have 



mp - k (fi - 1) + 1 ^ Oi + l) 2 - 0* - m. + k + l) 2 - 1 ; 



the vertex counting as k (ft — 1) points of intersection of C m and M^ . 

 We must therefore have in general 



(m - K -\f + ( K + 1) = 



^— < (i. 



m — k 



If the curve C m has in addition to this K-tuple point certain K'-tuple 

 points that are (k 1 + l)-tuple points on M^ , it is evident from reasoning 

 similar to that on page 506, that we must have 



(l ~ kOz ~ i) - 2(*' + 1)*+ 1 ^ [r> + 1) 2 - 2 (*+ ! ) 2 ] 



- [Qi - m + k + I) 2 - 2(# - K ' + X Y\ " l ' 



( m _ K _l)2 + ( K+ 1) _ 2k'(/c'-^'+.1) 



in 



i.e. 



m — k 



Zv> (I) 



where the summation extends over all multiple points of C m (except the 

 one at the vertex of K m _ K ) that are (k' + l)-tuple points on M M . The 

 smallest value of n that satisfies (I) is the order of the monoid that will 

 in this case cut C m out of K m — K . 



