WILLIAMS. — GEOMETRY ON RULED QUARTIC SURFACES. 23 



for which the formula holds. Now the formula holds for every genera- 

 tor, i.e. for a 1 , since 1 meets bp in (3 points, and the formula gives 

 (1 , bp) =■ 1./8 + b.O — fjL.0,/3 = /S. Therefore, if every conic can be cut 

 out by an SW such that the residual is uothing but generators, if every 

 cubic curve can be cut out by an S^ such that the residual consists 

 entirely of conies or generators or both, and in general, if every a a on 

 2^) can be cut out by an <SW such that the residual * is of order less 

 than a or is composed of curves of orders less than a, the formula is 

 true. 



5. For certain species of C^'s it may be possible to choose v smaller 

 than for certain other species, e. g. all the quartic curve* lying on a 

 quadric surface can certainly be cut out by cubic surfaces, but the 

 " quartics of the first kind " can also be cut out by quadric surfaces. 



We shall first determine the lowest value of v for which we can be 

 certain that an S^ will cut out any species of 0- a \ This can be done 

 for a surface of any order, fx, without difficulty, but since we are here 

 going to treat the ruled quartics only, we shall consider the case of 

 fx. — 4 only. 



iSW is determined by }.- (v + 1) (v + 2) (v + 3) — 1 arbitrary points. f 

 When v > 5 we must take care that SW does not break up into 2 ,4) and a 

 surface of order v — 4, i. e. of the points necessary to determine S^ we 

 must take one more than enough to determine a surface of order v — 4 as 

 not lying on 2 i4) ; also, we must take a v + 1 points of £<■') on C 1 " 1 in 

 order that SW may contain this curve ; so that for v > 5 the number of 

 arbitrary points of 2 ,4) through which we can make $» pass is 



(2) . . . £( v + l)( v +2)(v+3)-l_£(v-3)(v-2(v-l)-(av+l) 



= 2 v 2 - a 



V. 



For v — 4 we must take one point of »SW not lying on 2 (4) , but then the 

 term \ (y — 3) (v — 2) (v — 1 ) = 1 ; for v = 1, 2, or 3 we do not have 

 to take any points of »$W off 2 ,4 \ but then the term £ (v — 3) (v — 2) 

 (v — 1) = 0; therefore formula (2) gives the number of arbitrary 

 points for all values of v, when /x = 4. We have, therefore, 2 v 2 — a v > 0, 



which gives at once v >„; so that, for the lowest value of v, we have 



Li 



v =■ - when a is even, and v — when a is odd. In some cases it 



2 2 



has been found more convenient, and apparently necessary, to take v 



* The letter r, when used, shall always denote the order of the total residual. 

 t Salmon's Geom. of Three Dimensions, Chap. XI. 



