68 PROCEEDINGS OP THE AMERICAN ACADEMY. 



tic then has an apparent tac-node equivalent to two apparent double 

 points. 



The quartic that lies on the cone having a triple edge, is a " quartic of 

 the second kind ; " for, if it were a " quartic of the first kind," we could 

 pass a quadric through it and through the vertex ; the triple line would 

 then lie on the quadric and be a generator of one system ; the generator 

 of the other system that passes through the vertex would meet the cone 

 four times at the vertex and at least once on the curve, and would there- 

 fore coincide with an edge of the cone ; but an edge of the cone meets 

 the quartic once only, and therefore the quartic would be met by the 

 generators of one system of the quadric once only, and would not be a 

 " quartic of the first kind " as supposed. 



Every " quartic of the second kind " has three apparent double points 

 (or cusps), and cannot lie on a quartic cone with fewer than three double 

 or cuspidal edges ; two of these may unite, forming a tac-nodal edge, or 

 all three double edges may unite, forming a triple edge ; on the tac-nodal 

 edge the quartic has an apparent tac-node equivalent to two apparent 

 double points, and on the triple edge the quartic has an apparent triple 

 point equivalent to three apparent double points. Through every '"quartic 

 of the second kind " can be passed one and only one quadric, and, if the 

 quartic lies on a cone with a triple edge, the quadric always passes through 

 the vertex ; for, the triple line meets the quadric three times on the quar- 

 tic curve, and therefore lies on it, being a generator of one system of the 

 quadric ; the generator of the other system that passes through the ver- 

 tex coincides with an edge of the cone, as we have seen, and therefore 

 the residual consists of this edge and the triple edge. When the " quar- 

 tic of the second kind" lies on a cone with three double (or cuspidal) 

 edges, the quadric cannot go through the vertex (for if it did all tiiree 

 double or cuspidal edges would lie on it), and the residual is therefore 

 another " quartic of the second kind " because it has an apparent double 

 point (cusp) on each of the three double (cuspidal) edges ; the points of 

 crossing are the same as those of the original quartic, and the curves lie 

 on different sheets at these points. Now, the generators of the quadric 

 meet the cone four times, and, considering one system of generators, they 

 must meet the cone three times on one quartic and once on the other 

 quartic (since every '' quartic of the second kind " on a quadric meets 

 the generators of one system three times and those of the other system 

 once), i. e. on the quadric, one of the quartics is a 4,3, and the other is a 

 4i ; therefore, considered as lying on the quadric, these quartics inter- 

 sect in 10 points, (43, Aj) — A + 12 — 6 == 10. But on the cone each 



