52 PBOCEEDINGS OF THE AMERICAN ACADEMY. 



i.e. a — k < 4 a. 



If, now. we pass «i plane through any edge of any infinite set anil turn 

 it about BO thai it will not pass through any L'dge of any finite set. it 

 will cut out three other edges of the Infinite sets. Let p u p a , />,,, and p 4 

 he the uuinher of points of 6' 1 " 1 , respectively, on the four edges in this 

 j ilane ; we have then 



Px + Pi + p 3 + pt = a — k < 4 a, 

 and, since neither p u p.,, p g , nor p t can he less than a, each must be equal 

 to a and a — k = 4 a. Therefore, every edge of every infinite set meets 

 C 1 "' 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 C {a) is, as before, designated by the symbol 

 a a , 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), (a a , bp) = a ft + b a — 4 a ft, gives, 

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

 <i a and bp, exclusive of the number of their intersections at the vertex of 

 the cone. 



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

 number of branches of a a through the vertex, i.e. a a has an (a — 1 a )- 

 tuple point ;it the vertex. Let a a be the complete intersection of A"' 

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

 total number of intersections of a a and bp is the number of intersections 



of S u) and bp, which is by, and, since a = 4v, b v = — ; now, at the 



vertex, £(•> has a point of multiplicity (since a a is the complete 



intersection of >'" ' and the cone and has an (a — 4 u)-point at the vertex ). 

 and b ti has a (/> — 4 ft) -tuple point, so that b$ meets SW in 



^-^y b _ 4 ft ) = "''- (l ,e + ba-.iaft) 



point- at the vertex . sine' the total number of intersections of bp and 



> is - , the number of their intersections exclusive of those at the 



l 



vertex, i. e. the number of intersections of bp and </„ exclusive of those at 



the vertex, is f (afi + &a — l < i ft) — "ft+ ba — A a ft, 



which is the number given l>y formula (1). 

 Since the formula holds when <>., is the complete intersection of the 



cone and S'\ by Theorem III, it holds when "., i the partial intersec- 



