WILLIAMS. — GEOMETRY ON RULED QUARTIC SURFACES. 23 



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

 tor, i.e. for a Iq, since Iq meets b^ in (3 points, and the formula gives 

 (1q, b^) = 1.(3 + b.O — IX.0./3 — /3. Therefore, if every conic can be cut 

 out by an S^''^ such that the residual is nothing 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^ on 

 2('') can be cut out by an S'-''^ 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 curves 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 €("'>. This can be done 

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

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

 (1 = 4 only. 



S'^"^ is determined by J (v + 1) (i/ + 2) (i/ + 3) — 1 arbitrary points.f 

 When V ^ 5 we must take care that S^"') does not break up into 2'*' and a 

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

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

 not lying on 2'^' ; also, we must take a v + I points of S^") on C'"' in 

 order that S'-"^ may contain this curve ; so that for i/ > 5 the number of 

 arbitrary points of 2'*' through which we can make S^"^ pass is 



(2) . . . H'' + l)('^+2)(v+3)-l-^(v-3)(v-2(.-l)-(«i.+ l) 



= 2 u'^ — a V. 

 For V = 4 we must take one point of *S'M not lying on 2'^ but then the 

 term }^ (y - 3) (v — 2) {v - 1) = I ■ for v = 1, 2, or 3 we do not have 

 to take any points of S^"^ off 2'^', but then the term i {v — 3) (u — 2) 

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

 points for all values of v, when /a = 4. We have, therefore, 2 i'^ — a v ^ 0, 



.... _ a 



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



V = - when <i is even, and v = — v— when <i is odd. In some cases it 

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



* Tlie letter r, wlien used, sliall always denote tlie order of the total residual, 

 t Salmon's Gcom. of Three Diuiensioiis, Chap. XI. 



