530 PROCEEDINGS OF THE AMERICAN ACADEBIY. 



this septimic can have : 1) 0, 8 ; 2) 1, 9 ; and 3) 0, 9. The quadruple 

 point in this case is equivalent to three actual and three apparent double 

 points. The above three species can thus be obtained directly from 

 curves of kinds b"), a"), and a.'"), respectively ; no other species are 

 possible. 



4. A septimic curve with two triple points, the tangents at each of 

 which lie in one plane, can be obtained as the partial intersection of a 

 quartic cone having a triple edge and a quadric (monoid) that has a line 

 in common with the cone. The vertex is a point of multiplicity four on 

 the complete intersection and therefore, since the line passes through the 

 point, a point of multiplicity three on the septimic curve. As the point 

 is an ordinary point on the quadric, the three tangents to the curve at 

 the triple point lie in one plane. The other triple point is the point 

 distinct from the vertex where the sheet of the quadric crosses the triple 

 line. The quartic cone having a triple edge can have no double edge in 

 addition, due either to an actual or an apparent double point. The curve 

 has therefore just 7 • 3 — 3 • 4 or 9 apparent double points. This species 

 can be obtained directly from a) ; it is the only possible species. 



5. A septimic curve with two triple points, one of each kind, can be 

 obtained as the partial intersection of a sextic cone and a quartic monoid. 

 The sextic cone has an ordinary point of the curve as vertex and the 

 curve as base ; and the quartic monoid has the triple point at which the 

 tangents do not lie in one plane as a double point, and the line from it 

 to the vertex as an ordinary line of kind III. This quartic monoid is 

 determined by twenty arbitrary points in addition to its vertex and the 

 double point. In order to make this monoid contain the septimic that 

 has the vertex as an ordinary point and the double point as a triple 

 point, we must make it contain 4 • 7 — 3 — 2 • 3 + 1, or 20 additional 

 points on the curve. This is just the number at our disposal. The sex- 

 tic cone meets the inferior cone of the monoid in 18 lines, of which one 

 is the tangent line to the curve and 17 are lines common to the cone and 

 the monoid. The monoid has 12 lines on it. The line of kind III counts 

 as two lines of the monoid and as three lines common to cone and monoid. 

 There are thus 10 other lines on the monoid that must be made to count 

 for 14 lines common to cone and monoid. The sextic cone therefore has 

 4 double edges ; these are due to apparent double ^wints, as the curve 

 can have no actual double points in addition to the two triple points. As 

 the quartic cone cannot have more than 4 double edges in addition to 

 the 2 triple edges, it is evident that all lines of the monoid must lie 

 on the cone. The three branches of the curve at one triple point are cut 



