Geometry on the Cubic Scroll of the Plrst Kind. 5 



twisted curves, gave a representation of the curves on the 

 cubic scroll by an equation in two parameters; he considered 

 also the degree of the surface, which can be passed through 

 any curve on the general cubic surface, and the degree of 

 the residual intersection or complementary curve with the 

 number of arbitrary elements entering therein; and though 

 suggesting that this residual intersection is made up of 

 composite curves, some of which may be multiple, he did 

 not go further into their consideration. 



To these articles may be added one by Sturm ^) giving 

 a treatment of the properties of curves on the general cubic 

 surface and another by Rohn^) on the same subject, which 

 considers also the surfaces that can be passed through these 

 curves, and presents some theorems regarding- the residual 

 intersection on the general cubic surface. 



It is the purpose of the present paper to consider the 

 order of the surface of lowest order which can be passed 

 through any curve on the cubic scroll of the first kind; to 

 prove that the residual intersection may always be made up 

 of straight lines, with a consideration of the possible distri- 

 butions of the same (as was done for the case of the quadric 

 surface by Halphen^)); and also to treat in much greater 

 detail than has yet been done some of the further questions 

 of the geometry on this surface. 



') Ueber die Cnrven auf der allgemeinen Fläche dritter Ordnung, 

 Klein Ann., XXI, 457—515. 



^) Die Eaumcurven auf den Flächen dritter Ordnung. Leipz. 

 Ber., XLVI, 84—119. 



^) Mémoire sur les courbes gauches algébriques. C. K., LXX, 380. 



