58 PROCEEDINGS OF TIIE AMERICAN ACADEMY. 



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

 points. 



The quartic th.it lit* s on the cone having a triple edge, 18 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 j the triple line would 

 then lif 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 dm is at t he 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 inav 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 lias an apparent triple 

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

 of the second kind" can be passed one ami only one quadric, and. it' the 

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

 tin- 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, ami therefore 

 the residua] consists of this edge and the triple ed^e. When the " quar- 

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

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

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

 another " quartic of the second kind" because it lias 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 sheet-, at these points. Now, the generators of the quadric 

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



most meet the cone three times on one quartic and ono the other 



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

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

 once i, i. e. mi the quadric, one of the quartics i- a 1 . and the other is a 

 ■1,; therefore, considered as \\\wi on the quadric, these quartics inter 



■ in l" points, < 1 .. l,i \ -f 12 — 6 10. Bui on the cone each 



