Geometry on the Cubic Scroll of the First Kind. 19 



method we found n' = 2 as a minimum. Having then 

 n' = p — 2t , n^ = , we get 



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



k' = p — q — t, 



holding for all integral values of t from to ip. As 

 already seen, at t points of the double director the curve 

 (p,q) has a pair of branches lying one branch in either 

 sheet; at the p — ■ 2t other points where the curve (p , q) 

 intersects the double director there are single branches, and 

 at each of these points the generator at the point in the 

 other sheet occurs as a part of the residual intersection. 

 Hence in general here it must be that g' -j- h' -f 1' = P - — 2t, 

 gQ = h^ = l^ = 0; and no changes in the residual intersection 

 are possible without altering m'. The fact that every point 

 of the complete intersection lying on the double director is 

 a double point with a branch lying in either sheet is 

 evident at once since f' = 0. 



If t=:^p we have Ü at once a function of X" and jj," 

 giving 



I'. p> 2q, ^even^ m' = ^p ,f' = g' = h' = l' == ,k' = ip — q. 



Every point of this curve on the double director has 

 one or more pairs of branches passing through it, one branch 

 of each pair lying in either sheet. Here (o ^ 1 and hence 

 this is the surface of lowest order which can be passed 

 through any curve (p , q). This then gives also the only case 

 where with p even and p > 2q a (p , q) can be the complete 

 intersection of 2 and S, and that will occur here when and 

 only when p = 2q. 



In case p>2q or if p>2q and U ^ EJ^ f (X" , [jt''), it will 

 require a third surface S' to determine (p , q) without residual 



