54 PROCEEDINGS OP THE AMERICAN ACADEMY. 



at the vertex, aud if we pass the mouoid through 15 other poiuts of the 

 sextic it will contain the sextic. The residual will then consist entirely 

 of edges or multiple edges of the cone. 



In the same way it may be shown that all curves of order 7, and all 

 curves of orders 8, 9, and 10 for which a = 1, can be cut out by a quar- 

 tic monoid such that the residual will consist of edges or multiple edges 

 of the cone. 



When a ^ 2 (for which a ^ 8), or when a ^ 11, i. e. for all curves 

 not yet considered, we must take v > 5, where v is the order of the sur- 

 face of lowest order that cuts out a^ such that the residual coysists of 

 edges or multiple edges of the cone. We shall now show that such a 

 surface can be found for any cia, and the value of v given in terms of a 

 and a. Since the residual is to consist entirely of edges or multiple 

 edges of the cone, an arbitrary edge must meet the required surface in 

 V — a points at the vertex, i. e. the surface must have a (v — a) -tuple 

 point at the vertex of the cone. Let M^_^ be the required surface. 

 When V > 4 we must take care that M ' does not break up into the 

 quartic cone and a surface which must be an Mj_^_^, i. e. a surface 



which has a (v — a — 4)-tuple point at the vertex; Mj_^_^ can pass 

 through only 



K(i/-3)(i^-2)(v-l) - (i/-a-4)(v-a-3)(v-a-2)]-l 



arbitrary points, different from the vertex, and consequently if we make 



M _ pass through 



(A) ^[(v-3)(.-2)(.-l)-(./-a-4)(v-a-3)(v-a-2)] 



arbitrary points, not on the quartic cone, it cannot have the quartic cone 

 as a component ; the number of arbitrary points remaining, to determine 

 M , must be great enough to make it contain «„, which has a — 4a 

 branches through the vertex ; consequently we have 



(3) . . . av+\ - {a- 4a)(v - o) < I \.{^ + 1) (" + 2) (,. + 3) 



-(;^-a)(v-«+l)(v-a+2) - (v - 3) (v - 2) (v - 1) 

 + (i/ - a + 4) (,/ - a - 3) (i/ - a - 2)] - 1, 



from which we obtain the relation 



1 +aa+ 4av — 4a2^4av — 2u^+ 4v— 4a — 3, 

 i. e. 



,,, a(n!-2a+4) + 4 

 (4)... v^ ^ , 



