WILLIAMS. — GEOMETRY ON RULED QUARTIC SURFACES. 21 



a points, equal to the number in which it meets an arbitrary generator. 

 (If the cui've goes through a point common to all the generators, e. g. 

 the vertex of a cone, this point is not counted as one of the a points on 

 any generator.) This furnishes us a method for classifying all curves on 

 2^, and we shall use the symbol a a to denote a curve of order a that 

 meets each generator of the surface on which it lies in a points, a being 

 a constant. Similarly, bp will denote a curve of order b that meets 

 each generator j3 times. Each generator is itself a plane curve of order 

 1, and since it is not met by any other generator except those lying in its 

 plane, it is represented by 1 . In case 2^ has more than one system of 

 rulings, the lines of one system are chosen as the generators, with refer- 

 ence to which the classification is made for all curves on that surface, 

 and the lines of any other system are regarded simply as curves of 

 order 1. 



We shall use the symbol (a a , bp) to represent the number of intersec- 

 tions of any two curves a a and bp on the ruled surface. 



Professor Story * proved that for all curves lying on a quadric surface 

 (a a , bp) = a fi + b a — 2 a ft and for all curves lying on the cubic 

 scrolls (a a , bp) = a (3 + b a — 3 a ft and stated that it is probably true 

 that (a a , bp) = a (3 + b a — fx a ft for curves on a scroll of any order 

 p. It will be proved here that the formula is true for all ruled quartic 

 surfaces, i. e. that we have 



(1) (a,bp) =afi + bu — 4 aft 



and when we say that this formula holds for a certain curve we shall 

 mean that it gives the number of intersections of this curve with any 

 other curve on the scroll. It must be borne in mind that these formulae 

 for the cones of different orders give, in each case, the number of inter- 

 sections of the curves aside from those at the vertex, since the vertex is 

 not one of the a points on any edge. This will be proved for the 

 quartic cones. 



3. We shall first consider three general theorems,! which must be 

 proved before the formula can be established. The first may be called 

 the fundamental theorem and may be stated thus : 



Theorem I. — If a be the number of points of C (a) on an arbitrary gen- 

 erator, there are a points of C (rt) on each generator. 



* On the Number of Intersections of Curves Traced on a Scroll of any Order. 

 Johns Hopkins University Circulars, August, 1883. 



t These theorems were given by Dr. Story in his lectures, October to Decem- 

 ber, 1899. 



