56 PROCEEDINGS OF THE AMERICAN ACADEMY. 



5. We BhalJ qow consider the twisted quartics more in detail. Every 

 twisted quartic is a li. and, since it has at least two apparent double 

 points, the coi i which it liea 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 cuta <>nt the twisted 

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

 couut 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 lirst 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" lias two apparent double points, and the two double 

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

 and once 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 tip 

 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 ha- an additional double or cuspidal edge that meets the quadric 

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

 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 pa 

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

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

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



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

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

 cuspidal edges and the curve ha- two apparent cusps. When the quadric 

 that cuts out the quartic '_ r "es through the vertei the residual consists 

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



double point- or apparent CUSpS lie, but for every other quadric that 



passes through the "quartic of the first kind," the residual i- another 

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

 sliown as follows: sine,, 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 quanic; moreover, everj edge or multiple 



\. Dumber "f common i ■' it- superior and inferior cones 



Caylcy, < Collected Papi rs, V 



