WILLIAMS. — GEOMETRY ON RULED QUARTIC SURFACES. 49 



4. Thoisted Curves. — It may be shown in exactly the same way as 

 for the Quartic Scroll aS'(32-, 1), that formula (1) holds for every twisted 

 curve on the developable. 



Formula (1) holds, therefore, for every curve on the developable. 



X. Quartic Cones. 



1. Using the plane quartic curves as a base, we may say that there 

 are ten different species of quartic cones, corresponding to the ten species 

 of plane quartic curves.* But for our present purpose we need not 

 distinguish between double edges and cuspidal edges, and it will be 

 convenient to divide the cones into five groups, viz., (I) cones hav- 

 ing a triple edge, (II) cones having three edges double or cuspidal, 

 (III) cones having two edges double or cuspidal, (IV) cones with one 

 double or cuspidal edge, and (V) non-singular cones, i. e. cones with no 

 multiple edges. t 



A curve C(''), lying on a cone, has a i"-tuple point at the vertex, where 

 < ^, and any plane through the vertex meets C(") in k points there. 



2. Theorem I. — Every edge of a quartic cone meets C(") in the same 

 number of points, say a points (in addition to the k points at the vertex) ; 

 the number of points of C^") on a double edge is 2 a, and the number of 

 points of C(") on a triple edge is 3 a. 



Proof. — Consideriug group (I), if we pass a plane through the triple 

 edge and revolve it about this triple edge, it will cut out, in succession, 

 all the edges of the cone; the plane meets (7(") in k points at the ver- 

 tex and in a fixed number of points, say t points, on the triple edge, and, 

 as it meets (7(") altogether in a points, the same number of points, say a 

 points, lie on each edge of the cone. An arbitrary plane through the 

 vertex meets the cone in four edges, and therefore 



4 a + ^- = a, i. e. 4 a = a — ^. 



Taking the plane through the triple line, we have 



a -|- T = « — X; = 4 a, i. e. t = 3 a. 



Consider now group (II), and pass a quadric cone through the three 

 double or cuspidal edges and any chosen edge A of the quartic cone ; 

 these count for seven edges in the intersection of the two cones, and 



* Salmon's Iliglier Plane Curves, § 243. 



t When not qualified by the words double, cuspidal, triple, multijile, etc., the 

 word ed(/e will always moan a simple ed(je of the cone. 



VOL. XXXVI. — i 



