60 PROCEEDINGS OF THE AMERICAN ACADEMY. 



eut double points, and groups (III) and (IV), having six apparent 

 double points. Now, from an ordinary point on any curve of order m, 

 the number of apparent double points of the curve is h — ^?^ + 2,* where 

 h is the number of apparent double points of the curve from an arbitrary 

 point; therefore, since every twisted quintic on a quartic cone has one 

 branch through the vertex, the number of apparent double points from 

 the vertex, i. e. the number of double edges of the quartic cone, is 

 h — 3 > 1 ; moreover, since any twisted quintic can be cut out by a 

 cubic monoid, the six lines of the monoid that pass through the vertex 

 must count for seven, the order of the residual, and therefore the cone 

 has at least one double (or cuspidal) edge with which one of these six 

 lines coincides; therefore there is no quintic curve on the non-singular 

 cone. There is a twisted quintic, which is not a special case of any of 

 the groups given by Salmon, that has only three apparent double points, 

 but it has an actual triple point, where the three tangents do not lie in 

 the same plane, and the quartic cone on which it lies has a triple edge 

 due to this actual triple point. All the quartic cones, except the non- 

 singular cones, have species of twisted quiutics on them, and these may 

 be tabulated in the same way as the twisted quartics. 



We have seen that any twisted sextic can be cut out by a cubic monoid, 

 and, since the residual is of order six, the six lines of the monoid that 

 pass through the vertex may be six edges of the cone, and therefore the 

 non-singular cone, as well as each of the other cones, may have twisted 

 sextics on it. 



Any twisted curve of order 7 can be cut out by a quartic monoid which 

 has 12 lines that pass through the vertex, 9 of which may be edges of 

 the cone, forming the total residual, and therefore a quartic cone of any 

 group has on it some species of twisted curve of order 7. For a > 8 

 we have a > 2 for some species of a^, and therefore when a > 8, a quar- 

 tic cone of any group has on it some species of a^. 



* Salmon's Geom. of Three Dimensions, § 330, example 2. 



