20 PROCEEDINGS OF THE AMERICAN ACADEMY. 



ruled quartic surfaces, by means of a pencil of surfaces of order m and 

 groups of n generators in involution. In 1897, Rohn * treated some of 

 the properties of curves on the general quartic surface, considering also 

 the surfaces that can be passed through these curves and presenting some 

 theorems regarding the residual intersection. 



In 1883, Professor Story discovered a method by means of which he 

 was able to classify all curves lying on a quadric surface and to give a 

 formula for the number of intersections of these curves, thus obtaining, 

 by a synthetic process, the results already found by Cayley.t Professor 

 vStory applied his method to the cubic scrolls, classifying all curves lying 

 on these surfaces and obtaining a formula for the number of intersec- 

 tions of any two of these curves, analogous to that found for curves on 

 quadrics. 



By an extension of the analytical method of Cayley, Dr. Ferry % suc- 

 ceeded most admirably in treating analytically the " Cubic Scroll of the 

 First Kind," verifying the results of Professor Story for this surface, and 

 in a paper soon to be published, Dr. Ferry has also verified, in the same 

 way, Professor Story's results for the other cubic scroll. 



It is the purpose of the present paper to consider the classification of 

 curves on all ruled quartic surfaces ; to find the formula for the number 

 of intersections of any two curves that lie on the same ruled quartic sur- 

 face ; and to point out some of the most notable results obtained in the 

 course of the investigation. The equations of many of the ruled quartic 

 surfaces are so complicated that very serious difficulties arise when we 

 attempt to treat them analytically, and it has been found most convenient 

 to employ the synthetic method of Professor Story. 



2. For convenience, we shall use the symbol S'^"'^ to denote a surface 

 of order v, and S^*^' to denote a ruled surface of order jx. O^") will be 

 used to denote a curve of order a lying on the ruled surface in question. 

 By an arbitrary generator we shall mean any simple generator that bears 

 no special relation to the curve in question, e. g. in considering a plane 

 curve, it is any simple generator not lying in the plane of the curve. 

 It must be proved, first of all, that every curve C<") meets each gen- 

 erator of the surface on which it lies in a constant number of points, say 



* Die Raumcurven auf den Flaehen IVer Ordnung. Verliandlungen der K. 

 Sachs. Gesell. der Wiss. zu Leipzig, 1897. 



t On the curves situate on a surface of the second order. Coll. Math. Papers, 

 Vol. v., and Phil. Mag., 1861, pp. 35-38. 



X Geometry on the Cubic Scroll of the First Kind. Archiv for Mathematik og 

 Naturvidenskab, B. XXI. Nr. 3, 1899. 



