GEOMETRY ON RULED QUARTIC SURFACES. 



By Frank B. Williams, 

 Fellow in Mathematics, Clark University, Mass. 



Presented by William E. Story, May 9, 1900. 



I. Introduction. 



1. Ruled quartic surfaces have been studied systematically by Cayley, 

 Cremona, and Salmon, and to some extent by Rohn, Chasles, Schwartz, 

 Reye, Voss, Holgate, and others. Of the quartic scrolls, Cremona,* in 

 his excellent synthetic treatment, enumerates twelve species, while 

 Cayley,f in the most complete and masterly analytical treatment of 

 scrolls, divides these scrolls into ten species and gives a comparison of 

 his species with those of Cremona, stating that the latter's two remain- 

 ing species, though properly considered as distinct from the others, may 

 be regarded as sub-forms of his seventh and ninth species. As we shall 

 not have occasion here to take account of these sub-forms, we shall follow 

 Cayley's classification. The other ruled quartic surfaces are : the de- 

 velopable quartic, or torse, whose edge of regression is a twisted cubic, 

 and the quartic cones, which, although developable, must be considered 

 separately. The developable quartic has been quite thoroughly studied 

 by Cayley, Salmon, and others, but very little seems to have been done 

 with the quartic cones, which, as will appear later, present some very 

 interesting features. 



The consideration of curves in space has been the subject of a great 

 many articles by the most eminent geometricians, but curves on ruled 

 quartic surfaces in particular have received little attention. As early as 

 1861, Chasles $ gave a method of describing curves of order 4 m + n, on 



* Sulle superficie gobbe di quarto grado, Mem. della R. Istoria di Bologna, 

 Series II., T. VIII. 



t Second and Third Memoirs on Skew Surfaces, Otherwise Scrolls. Coll. 

 Math. Papers, Vol. V. and Vol. VI., respectively, and Phil. Trans., 1863 and 1869, 

 respectively. 



I Description des courbes a double courbure de tous les ordres sur les surfaces 

 reglees du troisieme et du quatrieme ordre. C. R. LIII, 884-889. 



