526 PROCEEDINGS OF THE AMERICAN ACADEMY. 



the quadruple point. It can be obtained from one of the remaining a)'s 

 only by having some of the apparent double points change to actual 

 double points in the limit. This can give no new species of sextic 

 curve. It is not possible to obtain the curve from any b), c), d), or e), 

 as there will not be enough apparent double points remaining. There is 

 thus only one species of sextic curve with a quadruple point and it has 

 four apparent double points. 



2. A sextic curve may have a triple point at which the tangents do 

 not lie in one plane. This sextic lies on a cone of order three that has 

 the triple point as vertex and the curve as base. This cone can have at 

 most one double edg^e. The curve can therefore have at most one actual 

 double point or one apparent double point when viewed from the triple 

 point. As the number of apparent double points of a sextic when viewed 

 from an arbitrary point in space is six more than when viewed from a 

 triple point of the curve it is evident that a sextic having such a triple 

 point may have either and 6, 1 and 6, or and 7. These sextics can 

 be cut out of the cubic cone by a cubic monoid. A cubic monoid is 

 determined by fifteen arbitrary points other than the vertex. In order 

 to make this monoid contain the curve we must make it contain nineteen 

 points of it. As the vertex, however, counts for six points common to 

 the curve and the monoid, it is only necessary to make the monoid con- 

 tain thirteen other points of the curve. There are thus two points left 

 at our disposal. The cubic cone meets the inferior cone of the monoid 

 in six lines, of which three are the tangent lines to the curve at the 

 triple point and three are lines common to the cone and the monoid. 

 These three lines and the sextic make up the complete intersection of 

 the two cubic surfaces. As the tangent lines at the triple point lie on 

 the quadric inferior cone of the monoid they do not lie in one plane. 

 As there are six lines on the monoid and only three lines common to 

 cone and monoid it is evident that it is not necessary for the cone to 

 have a double edge due to an apparent double point. If the cone and 

 the monoid have three distinct lines in common the curve has no 

 apparent double points when viewed from the triple point. It may or 

 may not have an actual double point in addition according as the cone 

 does or does not have a double edge. If the cone and the monoid have 

 only two lines in common, the curve must either have an apparent double 

 point when viewed from the triple point, or one of the lines common to 

 the cone and the monoid must be a line to an actual double point of the 

 curve. All three kinds above can therefore be obtained as the partial 

 intersection of a cubic cone by a cubic monoid. These three kinds can 



