VAN DER VRIES. — MULTIPLE POINTS OF TWISTED CURVES. 513 



points and one apparent double point. Tliis triple point does not have 

 its tangents lying in one plane. 



Consider now a curve of order m with an (m — 4)-tuple point and a 

 triple point. The curve can have no other actual multiple point. We 

 assume 6 < m. In this case we take the vertex of the cone at the triple 

 point, and must therefore know the reduction in the number of apparent 

 double points of a curve when it is viewed from a triple point of the 

 curve. We shall, however, consider the general case and consider the 

 reduction when the curve is viewed from a ^--tuple point on it. Plucker's 

 formulae * give 



K = 3,/ (i/-2) - 6T-8i, (1) 



3ju = 3v (v- 1) - 6t- 9^, (2) 



e = 3 iM (|M - 2) - 6 8 - 8 K, (3) 



K=Uf^(i«-l)-28-v]. (4) 



In the case we are considering, 



(I = m — k, V = r — 2 k, k =■ p. 

 Substituting these values in (1) and (2), and combining, we have 



I = yg — 3 (m — /(-■) + 3 (r - 2 k). 

 Substituting the same values in (3), we have 



i = 3 (m - k) (m-k-2)- G U, + ^ ^^V^) ~ 



8(3; 



where ^ extends to all multiple edges due to all multiple points other 



2 



than the vertex, and where 8i is the number of double edges due to 

 apparent double points. Eliminating i between these last two equations, 

 we have 



2 (Si + 2 ^-^^f-^) = ('" - ^y -m + 3k-r-3(3. (5) 



Now take the vertex of the cone at an arbitrary point in space. If 8 is 

 the number of double edges on this cone, we have 



* Salmon's Geometry of Tliree Dimensions (1882), No. 325, footnote. 

 VOL. xxxviii. — 33 



