Geometry on the Cubic Scroll of the First Kind. 39 



director where p — q = B -j- ''^ and B is the number of 

 branches of the corresponding plane curve at the point (x , y). 

 But this number of double points arises entirely from actual 

 double points where both branches lie in the same sheet, 

 and we have already seen that other actual double points 

 occur when two branches lying in different sheets meet at 

 the same point of the double director; the number of these 

 may be < ^p when a single branch in either sheet in each 

 case meets a single branch in the other sheet at a point of 

 the double director. But if a' branches in one sheet meet 

 at a point P^ on the double director and ß' branches in 

 the other sheet meet at the same point, these count as 

 i(a' -|-ßO(°^' + P' — 1) actual double points of (p,q); while in 

 the corresponding plane curve they count as |a'(a' — 1) -]- 

 ^ß'(ß' — 1) double points; hence the excess due to the 

 occurrence thus is ^ (a' -f ßO («' + ß' — 1) — \^' (P-' — 1) — 

 Iß' (ß' — 1) = a'ß' ; and therefore, if the p points of the 

 double director belonging to the curve (p , q) occur all together, 

 while half the branches lie in either sheet there, the curve 

 (p , q) will have |p" more double points than its analogue 

 in the plane; the upper limit in such a case is then 



ip' + p(q-i)-iq(q + i) + i. 



This fact that a curve (p , q) may have a greater or a 

 lesser number of double points than the corresponding plane 

 curve gives the second important distinction between the 

 geometry on 2 and plane geometry. 



