486 PROCEEDINGS OF THE AMERICAN ACADEMY. 



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

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

 1 (/A + 1) (/x + 2) (^ + 3) — J ^- (^ + 1) {k + 2) — 1 additional points. 

 The multiple point counts in general as p ^ intersections of Cm and 31^. 

 Therefore, in order to make S^j. contain Cm, it is only necessary for us to 

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

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



m^-pk^-l-^l (^+1) (/^ + 2) (^+3)-J^(^+l) (^^+2)-l. 



If, however, this inequality gives a value of fx 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 Si, having a ^^'-tuple point and a sur- 

 face Sfj.—v of order jx — v having a (k — ^•')-tuple point ; that is of the 

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

 surface S^—v must be taken as not lying on S^ This can be done if (i 

 satisfies the inequality : — 



^ {,x-v +1) {P.-V-V2) {,x-v + ?.) -l{k-k') (k-k' + \) (k-k' +2) 

 <Hi" + l) + 2) (/x + 3)-H-(^'+l) (/(,•+ 2)-l-(m/x-p^+l). 



Summing for all points that are p-tuple points on Cm while ^-tuple points 

 on Sfj. and ^-'-tuple points on Si,, we have fx. given in general by the 

 inequality : — 



^i«-2^^+l<n/^ + l.)0' + 2)(^ + 3)-U^(^+l)(i+2); (V) 



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



mix -^pk + 1 ^ J O + 1) (/. + 2) (/. + 3) - ^ ^ (^^ + 1) (^- + 2) 



_ 1 (^_ ^ + 1) (^ _ ^ + 2) (/. - V + 3) 



+ 1 {k-k'){k-k' ■\- \) ik - y -{■ 2), 

 that is by 



It is thus possible to find a surface aS> that will cut the curve from the 

 given surface S^,, where /a is the smallest integer that satisfies (V) or 

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

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



