VAN DER VRIES. — MULTIPLE POINTS OF TWISTED CURVES. 523 



tangent line in common the totality of the branches whose tangents lie in 

 one plane must be considered as having at least one connection or inter- 

 section with the totality of branches whose tangents lie in the other 

 planes. The whole system of tangent lines or of branches themselves 

 may thus in the limit be considered as forming a connected system. The 



difference between the number of actual double points and is 



the number of apparent double points contained in the composition of 

 the fc-tuple point. 



A (fc + l)-tuple point can be obtained by causing a new branch of the 

 curve to pass through a fc-tuple point of the curve. If this new branch 

 is added in such a way that its tangent lies in a plane with the tangents 

 to X of the branches through the fc-tuple point, it must be considered as 

 meeting the k branches together A times at the fc-tuple point ; and we 

 have : — 



Pk+i = Pk + A actual d. pts. -f (& — A) apparent d. pts. 



If the new branch is added in such a way that its tangent in the limit 

 lies in a planes that contain two other tangents to the curve at the fc- 

 tuple point, we must in general add three actual double points to the 

 value of the multiple point for each of these cr planes, viz. two for the in- 

 tersection of this new branch with the two whose tangents lie in the 

 same plane with its tangent and one for the intersection of these two 

 branches. This is shown if we consider a V (2) as obtained from a IV 

 (5). The new branch is added in such a way that its tangent lies in the 

 limit in three planes each of which contains two other tangents to the 

 curve ; nine actual double points thus being added to the value of a IV 

 (5) to obtain the value of a V (2). If, however, in the fc-tuple point y 

 connections were necessary involving branches whose tangents lie in 

 these o- planes, we must add only 3 o- — y actual double points. Thus, if 

 we consider a V (6) as obtained from a IV (8), we add only eight actual 

 double points, for the three branches through the sextuple point whose 

 tangents lie in one plane are considered as having one connection in the 

 limit with the branches whose tangents lie in the other plane. If the 

 new branch is added in such a way that its tangent lies in the limit in p 



planes that contain respectively X u A 2 , A 3 , A p other tangent lines, 



where 3 < A„, we must in general add 8 = A t -f A 2 + A 3 -f- + A p 



actual double points. If, however, in the limit there were a connections 

 necessary between these different sets of branches of the fc-tuple point, 

 we must add only 8 — a actual double points. Thus, if we consider a 



