508 PROCEEDINGS OF THE AMERICAN ACADEMY. 



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



in — K 



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



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 



7)1 — K 



and a surface of order u that has the vertex of the cone as a 



a 



(/J. — a)-tuple point ; every line through the (fi — 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 

 m — 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 Km~K in one point 

 distinct from the vertex. Such curves can be cut out of K^—k by some 

 monoid of order /u,. 



I. Suppose m — K ^ iJL. Then in order to make M^ contain Cm with- 

 out breaking up into Km—K and a monoid of order ^u — m + «, we must 

 have 



W// - K (iU - 1) + 1 ^ (/i + 1)' - (;u - m + K + 1)2 - 1 ; 



the vertex counting as k(/i — 1) points of intersection of Cm and My.. 

 We must therefore have in general 



(rn - K -If +(«+!)_ 



< |U. 



m — K 



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

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

 similar to that on page 506, that we must have 



mfx- k(j^ - 1) - 2^-' + 1)^-'+ 1 ^ [(f* + 1)' - 2 (^-'+1)'] 



- [(^ - m + K + 1)2 - ^(k> - k' + ly] - 1, 



(m - K - 1)2 + (k + 1) - 2 '<'('<' - ^' + 1) _ 



i. e. < fi ; (I) 



where the summation extends over all multiple points of Cm (except the 

 one at the vertex of i^m— «) t^at are (k' + l)-tuple points on M^. The 

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

 in this case cut C^ out of Km—g. • 



