VAN DER VRIES. — MULTIPLE POINTS OF TWISTED CURVES. 529 



four actual and six apparent double points. This septimic can be 

 obtained as the partial intersection of a quadric cone and a quartic 

 monoid, the residual intersection being a straight line. This curve has, 

 as we have seen, five apparent double points. It is thus obtainable 

 from a") above and is the only species of twisted septimic with a quin- 

 tuple point. 



2. A septimic curve can have a quadruple point at which three of the 

 tangents lie in one plane. Such a quadruple point is equivalent to four 

 actual and two apparent double points. This curve will lie on a cubic 

 cone that has the quadruple point of the curve as vertex and the curve 

 as base. If we wish to cut this curve out of the cubic cone by means of 

 a monoid, we must take a monoid that has the multiple point of the 

 curve as a point of multiplicity not less than three ; otherwise the three 

 tangents cannot lie in one plane. The monoid must therefore be of an 

 order as great as four. A quartic monoid is determined by 24 points. 

 Four of these points must be taken off the cubic cone to insure the 

 monoid not breaking up into this cone and a plane. In order to make 

 this quartic monoid contain the septimic curve that has the vertex of the 

 monoid as a quadruple point, we must make it contain 4 • 7 — 3 • 4 — 1 

 additional points of the curve. There are thus still three points left at 

 our disposal. The cubic cone meets the hiferior cone of the monoid in 

 nine lines, of which four are tangent lines to the curve at the quadruple 

 point and five are lines common to the two surfaces. It is evident, as in 

 previous cases, that either four or five of the lines of the monoid are 

 edges of the cubic cone, this cone having at most one double edge. 

 Septimic curves with quadruple points of the above kind are thus of 

 three kinds, viz., 1) 0, 8 ; 2) 1,8; and 3) 0,9. These can be obtained 

 directly from b'), a'), and a") above ; there are no other possible cases. 



3. A septimic curve with a quadruple point at which no three tangents 

 lie in one plane can be obtained as the partial intersection of a cubic cone 

 and a cubic monoid. The cubic cone has the quadruple point of the 

 curve as vertex and the curve as base. The cubic monoid is determined 

 by 15 arbitrary points. In order to make it contain a septimic curve 

 that has the vertex of the monoid as a quadruple point, we must make it 

 contain 3*7 — 2*4+1 or 14 additional points of the curve. Tiiere is 

 thus one point left at our disposal. The cubic cone meets the inferior 

 cone of the monoid in six lines, of which four are tangents to the curve 

 at the quadruple point and two are lines common to the cone and the 

 monoid. These two lines and the septimic make up the complete inter- 

 section of the two surfaces. As in the previous case, it is evident that 



VOL. XXXVIII. — 34 



