486 PROCEEDINGS OP THE AMERICAN ACADEMY. 



In the case we are concerned with in particular, the surface S^ has a 

 £-tuple poiut at a p-tuple point of the curve. S^ is then determined by 

 £ (p. + 1) (ji + 2) Qi + 3) — %k (k + 1) (* + 2) — 1 additional points. 

 The multiple point counts in general as p k intersections of C m and M^. 

 Therefore, in order to make S^ contain C m , it is only necessary for us to 

 make it contain m p. — pk -\- 1 points of the curve in addition to the 

 multiple point. Therefore p must in general satisfy the inequality : — 



m fi - p h + 1 ^ J- (/* + 1) 0* + 2) 0* + 3) - \h Qe + 1) (k + 2) - 1 . 



If, however, this inequality gives a value of p. that is greater than the 

 value of v, care must be taken that the surface S^ having a &-tuple point 

 does not break up into the surface S„ having a &'-tuple point and a sur- 

 face Sn-v of order p. — v having a (k — &')-tuple point ; that is of the 

 points necessary to determine S^ one more than enough to determine the 

 surface S^—,, must be taken as not lying on S v . This can*be done if p 

 satisfies the inequality : — 



J0»-v+l)0t-v+2)0t"V + S)-J(*-* r )(*-f+l)(*-* / +2) 

 ^i0» + l)0* + 2)0* + 8)- J* (t+l)(*+2)-l-(«i /!-/»*+ 1). 



Summing for all points that are p-tuple points on C m while &-tuple points 

 on S^ and i'-tuple points on S y , we have p, given in general by the 

 inequality : — 



mp-^pk+\^l{p+\){p + 2){p + Z)-lk{k+\){k + 2)', (V) 

 or, if this gives a value of p such that v < p, by the inequality : — 



»j*-2p*+ 1 <*-<' l + 1 >G t + 2 > (** + «)-**(*+ !)(*+*) 



- i (p, - v + 1 ) 0* - v + 2 ) O - v + 3) 



+ i (k-V)Xk-V+l) (Jc-V + 2), 

 that is by 



sf 



>_ m 0_t)^ + l + i + S^l(*+l)(4-*:+l)+i^+l)(*'-l)l-2Sc; 



It is thus possible to find a surface S^ that will cut the curve from the 

 given surface £„, where p. is the smallest integer that satisfies (V) or 

 (VI) ; using (V) if it gives a value p. < v, otherwise using (VI). It 

 may be possible to cut the curve out of S v by a surface of lower order 



