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



points and one apparent double point. This 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 = 3v(v — 2) — 6t — 8«, (1) 



3^i = 3v 0- 1) - 6 T - 9», (2) 



i = 3 n - 2) - 6 S - 8 k, (3) 



k=^ 0(f-l)-2S-v]. (4) 



In the case we are considering, 



(i = m — k, v = r — 2 k, k = /?. 

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



* = j3 — 3 (m — k) + 3 (r — 2 k). 

 Substituting the same values in (3), we have 



i = 3(m-k)(m-k-2)-6 (s, + ^ p( \ l) \ 



8/3; 



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



2 



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

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

 we have 





P( - P ] M = (m - k) 2 - m + 3 k - 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 Three Dimensions (1882), No. 325, footnote. 

 vol. xxxvm. — 33 



