WILLIAMS. — GEOxMETRY ON RULED QUARTIC SURFACES. 51 



turn it about so that it will not contain any other edge of any finite set, 

 and this plane will then cut out three u-edges in addition to the chosen 

 edge ; if )8 is the number of points of (7*°' on the chosen etlge, we have 

 (3 -\- 3 a =^ a — /t = 4 a, i. e. y8 = a. 



Therefore every edge of the cone has a points of O"^ on it. If the cone 

 has a double or cuspidal edge, an arbitrary plane through it will cut out 

 two a-edges, and if 8 is the number of points of C'"' on the double or 

 cuspidal edge 



8 + 2 a =: a — Z: = 4 a, i. e. 8 = 2 a. 



• 



If the cone has a triple edge, having t points of C'"' on it, an arbitrary 

 plane through this triple edge will cut out one a-edge, and we have 

 T + a = a — k = 4 a, * i. e. T = 3 a. 



Therefore, the theorem holds when there is only one infinite set. Sup- 

 pose now that any number of the a sets are infinite, and let a be the 

 least value of p belonging to any of these infinite sets. Pass a cubic 

 cone through nine of these a-edges; then, since the cubic cone meets 

 C'"' in 3 a points, of which 3 ^- lie at the vertex and 9 a on the nine 

 chosen edges, the remaining 3a — 3k — 9a points lie on the three 

 other edges in which the cubic and quartic cones intersect, and at least 

 one of these three edges must have as many as a — k — 3 a points of 

 C"' on it. Keeping seven of the a-edges fixed, we can vary the other 

 two in such a way that the cubic cone, determined each time by the nine 

 a-edges, will have for its remaining intersection with the quartic cone 

 tliree edges different from those of any other such cone previously deter- 

 mined ; since there is an infinite number of a-edges, we get, in this way, 

 an infinite number of such cubic cones, and therefore an infinite number 

 of edges each having as many as « — ^' — 3 a points of C'"' on it ; hav- 

 ing an infinite number of such edges, we can so choose two of them that 

 their plane will not pass through any edge of any finite set, because there 

 is a finite number of edges in all the finite sets taken together ; besides 

 the chosen edges this plane will then cut out two edges belonging to the 

 infinite sets. Now the number of points of (7'"* on the two chosen edges 

 taken together is equal to or greater than 2a — 2^— 6 a, and if (3 and 

 y be the number of points of C'-"\ respectively, on the other two edges 

 in the plane we must have 



2 a — 2k-C>a + l3 + y^a — k, 



and, since neither (3 nor y can be less than a, 



2a-2^-— 6a-t-2a^'^-/C', 



