VAN DER VRIES. — MULTIPLE POINTS OF TWISTED CURVES. 531 



out of the monoid at the double point by the three sheets of the cone, 

 and thus do not have their tangents lying in one plane. The three 

 branches of the curve at the other triple point are cut out of the cone by 

 an ordinary sheet of the monoid at the point where the sheet crosses the 

 second triple line ; the tangents at this point thus lie in one plane. This 

 curve has 4 apparent double points when viewed from an ordinary point 

 of the curve, or 9 apparent double points when viewed from an arbitrary 

 point. It can be obtained directly from a') above. 



6. A septimic curve that has two triple points of the second kind can 

 be obtained as the partial intersection of a quartic cone and a cubic 

 monoid. The quartic cone must have one of the triple points as vertex 

 and the curve as base. The cubic monoid must have the second triple 

 point as a double point and the line from it to the vertex as an ordinary 

 line of kind III. This monoid is determined by 11 points in addition to 

 the vertex and the other double point. In order to make this monoid 

 contain a septimic curve that has each of the double points of the monoid 

 as a triple point, we must make it contain 3*7 — 2*3 — 2*34-1 or 10 

 additional points of the curve. There is thus one point left at our dis- 

 posal. The quartic cone meets the inferior cone of the monoid in 8 

 lines, of which 3 are tangent lines to the curve at the triple point at the 

 vertex and 5 are lines common to the cone and the monoid. The quartic 

 cone can have no double edge in addition to the triple edge to the second 

 triple point. The curve has therefore no apparent double points when 

 viewed from one of the triple points*; it therefore has just 9 when 

 viewed from an arbitrary point in space. This curve can be obtained 

 from a septimic of kind a"), the two triple points counting as 4 actual and 

 2 apparent double points. It is to be noticed that when we wish to ob- 

 tain a curve having a multiple point from a curve that has only double 

 points, we cannot always use the curve that has the minimum number of 

 apparent double points. 



7. A quartic cone and a quadric (monoid) that have a line in common 

 intersect in addition in a septimic curve that has a triple point at the 

 vertex of the quartic cone. The tangents to the curve at the triple point 

 lie in the tangent plane to the quadric at this point. The quartic cone 

 may have three double edges, but none due to an appareut double point, 

 for such an edge would count twice as a line common to cone and monoid. 

 The curve therefore has 7 * 3 — 3 * 4 or 9 apparent double points. This 

 is also evident from our formula : — 



h + h' + h" = ^o<-i)(--i) _ *y(fc-i)(fr-i) t 



- u 



