WILLIAMS. — GEOMETRY ON RULED QUARTIC SURFACES. 25 



C ia> with au arbitrary plane through the line L lie on Z), we can then 

 make L lie on the surface *SW that cuts out C [a) and still have left at our 

 disposal the number of points given by formula (2). For, if the v+1 

 points of C (a) on L are all ordinary points, L meets $» v+1 times and 

 therefore lies on $"> ; but, if C (c ° has an ra-tuple point and v — m + 1 

 other points on L, even if L meets <$» only once at this multiple point, 

 m — 1 points not included in formula (2) are still at our disposal and 

 may be taken on L, so that v+1 points of <SM lie on L and L will 

 therefore lie on -SW. Therefore, in considering whether a line can be 

 made to lie on S^ or not, we need not take account of the multiple 

 points of C {a) that lie on this line, but may regard the line as meeting 

 £("> in points of C {,,) equal in number to the number of points on the 

 line in which an arbitrary plane through the line meets C" a) . 



6. We have seen that, when certain theorems can be proved, formula 

 (1), p. 21, gives the number of intersections of any two curves on the 

 same ruled surface. In special cases, where the curves bear a particular 

 relation to one another, and in most cases where the multiple curve is 

 involved, the result given by this formula requires a special interpreta- 

 tion, namely : if two curves on 2 (41 pass through the same point of the 

 multiple curve, any branch of either curve is regarded as intersecting 

 only those branches of the other curve that lie on the same sheet with it, 

 and two branches that pass through the same point of the multiple curve 

 are not regarded as intersecting at that point if they lie on different 

 sheets of 2 (4) there. In particular, two generators through the same 

 point of the multiple curve are not regarded as intersecting, when con- 

 sidered as loci on that quartic surface. 



The double curve on a ruled quartic cannot be of order greater than 

 three, and therefore a plane section can never have more than three 

 double points on the double curve. If the plane section has a double 

 point not on the double curve, this double point is a point of tangency of 

 the plane, and, since a tangent plane to a ruled surface contains the gen- 

 erator through the point of tangency, the section must be a degenerate 

 quartic curve having at least one generator as a component. Therefore 

 the section of a ruled quartic by a plane can never consist of two proper 

 conies ; for the section would then have four double points, one of which 

 must be a point of tangejicy of the plane, and therefore the plane would 

 cut out a generator. 



Cayley * uses the general symbol £ (m, n, p) to denote a scroll gener- 



* Second Memoir on Skew Surfaces, Otherwise Scrolls. Coll. Math. Papers, 

 Vol. V., and Phil. Trans., 1863 



