506 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



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

 tains in general (jU + 1)^ — (^' 4-1)^—1 arbitrary consonants. In 

 order to make this monoid i^ contain a curve Cm that has the vertex 

 oi Mfj, as an ordinary point and a (^ + l)-tuple point of il/^ as a K-tuple 

 point, we must make it contain m/< — (/i — 1) — « (/; + 1) + 1 addi- 

 tional points of Cm. ' This is possible if 



mti-{^-\)-K{h + i) + \^{^ + \y-(k+ \y-i, 



m — 3 



i.e. if 



+ i \^m^ — 6 w + 17 — 4 (^ + 1) (k — ^ - 1) ^ ^ ; 



or, summing for all »c-tuple points of Cm that are (Jc -f- l)-tuple points 

 on M^ , we must have 



m 



- + i ^ni" - ^m + n - 4.^{k + \){k- k-1) Z fi. (I) 



The smallest value of |^ that satisfies (I) will in general suffice as the 

 order of a monoid that can be made to cut Cm out of Km—\ - As 

 {k -\- 1) (k — k — 1) is never less than 0, it is evident that ^i = 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 ^ can never be less than the smallest integer that satisfies 



m — 1 



+ i i/"^^- &ni -]- 1 + S ^ k {2 k - K + 2) + S a- ^ ft. (II) 



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



A curve of order . . . . ni 



having a point (or points) of 

 multiplicity k 



can be obtained as the 

 intersection of a cone 

 of order m — 1 



and a monoid of order . . jW 



tliat has the K-tuple point 

 (orpoints) of the curve 

 as points of multiplic- 

 ity k+ I 



3,3 



6 

 4 



2,2 



4,3 



7 

 4 



2,2 



3,3 



7 

 5 



2,2 



