VAN DER VRIES. — MULTIPLE POINTS OF TWISTED CURVES. 527 



be obtained directly from b'), a'), and a") above. The other cases either 

 give us these in the limit, or give a curve with more actual double jjoints 

 or fewer apparent double points than the sextic curve can or must have. 



The sextic with a double point in addition to the triple point can be 

 obtained as the partial intersection of a quartic cone, having the double 

 point as vertex and the curve as base, and a cubic monoid having the 

 triple point of the curve as a double point and the line from it to 

 the vertex as an ordinary line of kind III. The quartic cone meets the 

 inferior cone of the monoid in eight lines, of which two are tangent lines 

 at the double point and six are lines common to cone and monoid. The 

 line to the triple point counts as three of the six lines of the monoid. 

 The cone need therefore have no double edges, and in fact cannot have, 

 as it already has a triple edge. Three lines in addition to the line to the 

 triple point must therefore be common to cone and monoid. The three 

 branches of the curve at the triple point are cut out of the monoid at the 

 double point by the three sheets of the cone that pass through the triple 

 line. The tangent lines at the triple point are the tliree lines in which, 

 in addition to the line from the triple point to the vertex three times, the 

 tangent planes to the three sheets of the cone intersect the tangent cone 

 at the double point of the monoid. As the curve can have no apparent 

 double points when viewed from the double point, it has 2 m — 6 or 6 

 apparent double points, when viewed from an arbitrary point in space. 



3. A sextic curve can have a triple point at which the tangents lie in 

 one plane. Such a sextic can be obtained as the intersection of a cubic 

 . cone and a quadric (monoid) that passes through the vertex of the cone 

 but has no line in common with it. The vertex, being a triple point on 

 the cone and an ordinary point on the monoid, is a triple point on the 

 curve. The tangents at this point lie in one plane, for they are the inter- 

 sections of the cubic cone by the tangent plane to the quadric at this 

 point. The cone can have no double edge due to an apparent double 

 point of the curve, for this would cause the sextic to break up into a 

 quartic curve and the line doubled. It may, however, have a double 

 edge due to an actual double point of the curve. The curve, whether it 

 has an actual dou])le point or not, has therefore no apparent double 

 points when viewed from the triple point. We thus obtain two species 

 of sextics with such a triple point, viz., 1) and G, and 2) 1 and G. 



A sextic curve with such a triple point can also be obtained as the 

 partial intersection of a quintic cone and a cubic monoid. The quintic 

 cone has an ordinary point of the curve as vertex and the line from this 

 point to the triple point of the curve as a triple edge. The monoid is 



