56 PROCEEDINGS OF THE AMERICAN ACADEMY. 



5. We shall now consider the "twisted quartics more in detail. Every 

 twisted quartic is a 4i, and, since it has at least two apparent donble 

 points, the cone on which it lies must have at least two double or cus- 

 pidal edges, i. e. no twisted quartic can lie on the cones of groups (IV) 

 and (V) ; moreover, the cubic monoid, which cuts out the twisted 

 quartic, has six * lines on it through the vertex, and these six lines must 

 count for eight, the order of the residual intersection of the cone and 

 monoid, and therefore the cone must have, at least, two double or cus- 

 pidal edges with which two of the six lines coincide. Through a "quar- 

 tic of the first kind" can be passed an infinity of quadrics, i.e. we can 

 pass a quadric through a " quartic of the first kind " and any arbitrary 

 point ; let this quadric be passed through the vertex. The " quartic of 

 the first kind " has two apparent double points, and the two double 

 edges, of the cone, on which they lie, meet the quadric twice on the curve 

 and ouce at the vertex, and therefore lie entirely on it ; these double 

 edges are, therefore, the two generators of opposite systems, of the 

 quadric, through the vertex. The cubic monoid, in this case, breaks up 

 into the quadric and a plane through the vertex. The " quartic of the 

 first kind" may have an actual double point or cusp, in which case the 

 cone has an additional double or cuspidal edge that meets the quadric 

 only once at this double point or cusp and once at the vertex, and there- 

 fore does not lie on it. The cone may have a cuspidal edge due to an 

 apparent cusp on the quartic curve, i. e. if a tangent to the curve passes 

 through the vertex, the curve when viewed from the vertex appears to 

 have a cusp on tliis tangent, which is therefore a cuspidal edge of the 

 cone (the apparent cusp replaces one of the apparent double points and 

 the cuspidal edge is one of the generators of the quadric). If two tan- 

 gents to the quartic curve pass through the vertex, the cone has two 

 cuspidal edges and the curve has two apparent cusps. When the quadric 

 that cuts out the quartic goes through the vertex the residual consists 

 entirely of the two double or cuspidal edges on which the apparent 

 double points or apparent cusps lie, but for every other quadric that 

 passes through the " quartic of the first kind," the residual is another 

 " quartic of the first kind " similar to the original quartic ; this may be 

 shown as follows: since the quadric does not go through the vertex no 

 edge or multiple can lie on it, and therefore the residual cannot break 

 up, i. e. it must be a twisted quartic; moreover, every edge or multiple 



* Tliis is the number of common edges of its superior and inferior cones- 

 Cay ley, Collected Papers, V. p. 8. 



