ARGHIV FOR MATHEMATIK OG NATÜRYIDEN8KAB. B. XXI. Nr. 3. 



Geometry on the Cubic Scpoll of the First Kind 



By 



Frederick C. Ferey, 



Fellow in Mathematics, Clark University. 



I. Introductory. 



The consideration of the curves which lie on the ruled 

 cubic surfaces has been, in whole or in part, the subject of 

 many articles pubhshed in the mathematical journals. As 

 early as the year 1861 Chasles^) gave a method of describing 

 curves of order 3m -|- n on the cubic scroll by means of a 

 pencil of surfaces of order m and groups of n generators 

 in involution. In 1866 Clebsch^) gave a treatment of the 

 geometry on the general cubic surface, which was somewhat 

 analogous to the study of the curves on the hyperboloid 

 already made in 1861 by Chasles^), and included an account 

 of the curves cut from the cubic surface by surfaces of the 



^) Description des courbes à double courbure de tous les ordi-es 

 sur les surfaces réglées du troisième et du quatrième ordre. C. E., 

 Lin, 884—889. 



^) Die Geometrie auf den Flächen dritter Ordnung. Kronecker J., 

 LXV, 359—380. 



•') Theorie analytique des courbes à double courbure de tous les 

 ordres tracées sur l'hyperboloïde à une nappe. C. R., LIII, 985 — 996. 



