i06 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



equation of a monoid of order /* that has a £-tuple line of kind III con- 

 tains in general (^ + l) 2 — (k + I) 2 — 1 arbitrary consonants. In 

 order to make this monoid M^ contain a curve C m that has the vertex 

 of M^ as an ordinary point and a (k + l)-tuple point of M^ as a K-tuple 

 point, we must make it contain m ,p — (ji — 1) — k (k + 1) + 1 addi- 

 tional points of G m . This is possible if 



m ll -(j l -l)-K(k+ 1) + 1^ (> + l) 2 -(^+ 1) 2 -1, 

 i. e . if W -^- 3 + i Vw» a - 6 m + 17 - 4 (k + 1) (k - k - 1) ^ p ; 



or, summing for all K-tuple points of C m that are (k + l)-tuple points 

 on Mn , we must have 



m — 3 



- + 1|/™ 2 - 6m +17-42(*+ !)(-<- *- !)</"• 



(I) 



The smallest value of /* that satisfies (I) will in general suffice as the 

 order of a monoid that can be made to cut C m out of K m —\ . As 

 {k -f 1) (k — yfc — 1) is never less than 0, it is evident that p = m — 2 

 will always suffice. As in the previous section, k can take any value 



from to or - — inclusive. We can show in the same way as 



there that p can never be less than the smallest integer that satisfies 



m — 1 



+ I i/m 2 - 6 m + 1 + 8 ^ k ( 2 k ~ K + 2 ) + 8 °" < /*• 



(H) 



Using (I) and (II), we can tabulate as follows : — 



A curve of order 



m 



having a point (or points) of 



multiplicity k z=z 



can be obtained as the 

 intersection of a cone 

 of order m — l = 



and a monoid of order . . H =. 



that has the K-tuple point 

 (or points) of the curve 



as points of raultiplic- 



ity k + 1 = 



