52 PROCEEDINGS OF THE AMERICAN ACADEMY. 



i. e. a — ^' ^ 4 a. 



If, now, we pass a plane through any edge of any infinite set and turn 

 it about so that it will not pass through any edge of any finite set, it 

 will cut out three other edges of the infinite sets. Let pi, p^, p^, and JO4 

 be the number of points of C'*"', respectively, on the four edges in this 

 plane ; we have then 



/>i 4- /'2 + />3 + i»4 = a — j{; < 4 a, 



and, since neither pi, p^, ps, nor p^ can be less than a, each must be equal 

 to a ^nd a — ^ = 4 a. Therefore, every edge of every infinite set meets 

 C'"' in a points, i. e. there is only one infinitive set, and we have proved 

 that the theorem holds in this case. 



3. On the cones, a curve CI"' is, as before, designated by the symbol 

 fla, but a now means the number of points, other than those at the 

 vertex, in which an arbitrary edge of the cone meets the curve C'"'. We 

 shall now show that formula (1), {(1^-, b^) = a (3 + b a — 4 a ft, gives, 

 for the quartic cones, the number of intersections of the two curves, 

 Oa and bp, exclusive of the number of their intersections at the vertex of 

 the cone. 



We have seen that a — ^- = 4 a, i. e. ^' = a — 4 a, where k is the 



number of branches of fta through the vertex, i. e. «„ has an (a — 4 a)- 



tuple point at the vertex. Let »„ be the complete intersection of S^''") 



and the quartic cone, and let b^ be an arbitrary curve on the cone ; the 



total number of intersections of «„ and b^ is the number of intersections 



a b 

 of /S'"^ and b^, which is b v, and, since a = 4 i', 61/ = — - ; now, at the 



vertex, /S^") has a point of multiplicity (since eta. is the complete 



intersection of S^") and the cone and has an (a — 4 a)-point at the vertex), 

 and b^ has a (6 — 4 ^) -tuple point, so that b^ meets /S^") in 



(^°)(*-*»=^*-(«fl+*a-4a/3) 



points at the vertex ; since the total number of intersections of b^ and 



/S'W is — , the number of their intersections exclusive of those at the 

 4 



vertex, i. e. the number of intersections of b^ and «„ exclusive of those at 



the vertex, is— + (a ft + b a — 4: a (S) = o/3+ b a — 4 a /3, 



4 4 



which is the number given by formula (1). 



Since the formula holds when «„ is the complete intersection of the 

 cone and S^''\ by Theorem III, it holds when a^ is the partial intersec- 



