54 PROCEEDINGS OF THE AMERICAN ACADEMY. 



at the vertex, and if we pass tbe monoid through 15 other points of the 

 Bextic 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 Bhown that all curves of order 7. and all 

 curves of orders 8, '.', and 10 for which a = 1, can he 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 Bur- 

 of lowest order that cuts out <i n such that the residual consists of 

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

 surface can he found for any o a , and the value of v given in terms of " 

 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 3P _ a be the required surface. 

 "When v > 4 we must take care that w ' does not break up into the 



^ v — a * 



• juartic cone and a surface which must be an -fl/„_ a _ 4 , Le. a surface 



which has a {v — a — l)-tuple point at the vertex ; Mj~ a _^ can pass 

 through only 



i [(V _ 3) - 2) (c - 1 ) - (v - a - 4) (v - a - 3) (v - a - 2) j - 1 



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



M _ ( pass through 



I £ [(„ _ 3) ( v _ 2) (v- 1)- (v- o- 4) (v- a- 3) (v- a- 2)] 



arbitrary points, not on the quartic cone, it cannol have the qnartic cone 

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

 .1/ ' . must be greal enough to make it contain a» . which has a — 4a 

 branches through the vertex • consequently we have 



(3) . . . a v + 1 - (a - 4a)(v - a) < \ [(»< + 1) (v + 2) (v + 8 1 



_(v-a)(v-a • l)(v-a + 2) - (. - -1) 



v — a-\ h n - a - 3) (v - a - 2)] - 1. 

 from which we obtain the relation 



1 + u a -t 1 a i I - 1 " i -2 -i- + 4 i> — \ a — 3, 



i. e. 



a (a - 2.. -* I • 1 



1) . . i . 



