22 Frederick C. Ferry. 



but since this gives a value of m' no lower than that of II 

 it is not of so great interest as is I'. 



Since no curve here is the complete intersection of 2 

 and S it will always be necessary to add a third surface S' 

 in order to completely determine (p , q) as an intersection of 

 surfaces. To determine S' for that purpose we have 



S; m'=p-t,f' = 0,g'-Ug^H-h' + h^ + l' + l^ = p-2t, 



k' = p — q — t, 

 S';m" = i(p + l),f"=l,g" + h" + l" = 0, 



k" = l(p-l)-q, 



which ansAver if t = p — q or p=i2q-|-l; otherwise we 

 must have 



S; m'=p-t,f' = 0,g' + g^ + h' + m-l'-f-l^ = p-2t, 



k' = p — q — t , 

 JS'; m" = p — q,f"=.p — 2q,g" + h"-{-l" = 0,k" = 0. 



Thus determined, 2 = , S == , and S' = form a 

 restricted system for the curve (p , q) . 



3. If p<2q, the conditions of p. 17 are satisfied by 

 n^ = 1 and n' = 2q — p -j- 2 , giving 



III. p<2q,m'=^q4-l,f' = l, 



g ^ + g^ + h^ + b^ + l^ + l^ = 2q-p + l,k^ = 0. 



Here w^to^v, where (u^ is an arbitrary homogeneous 

 polynomial in \ and [i of degree 2q — p + 1 j SLud hence the 

 portion of the residual intersection represented by g' -j- g^ + 

 h' -I- hp -f 1' + Iq ^^ ^*1 — P "h 1 J^^äy be made up of any single 



