26 PROCEEDINGS OF THE AMERICAN ACADEMY. 



1 by a line that meets each of three curves of orders m } n, and p once, 

 .-.m to denote a scroll generated by a line that meets a curve of 

 order m twice and a curve of order n once, and S (m 9 ) to denote :i scroll 

 g rated by a line that meets a curve of order /// three times. In his 

 Bymbols for the quartic scrolls be has also used a subscript, in most cas< -. 

 to denote the order of multiplicity of the curve on the scroll : but he lias 

 not, in all cases, adhered to his general method, and it senus best, while 

 preserving his classification, to change his symbols, making them con- 

 form to his general rule for such symbols. 



II. Quartic Scroll, with a Triple Linear Director and a 

 Simple Linear Director, S(\ 3 , 1, 4). (Catley's Third Sfecu s, 

 S(l 8 ,l,4).) 



1. This scroll has three sheets through the triple linear director, 

 which we shall denote by T, and T is scrolar* on each sic 



Through each point of T pass three generators, one on each sheet, 

 and if we pass a plane through two of these generators it will also con- 

 tain the simple director, since each generator meets the simple director 

 once, and therefore the third generator at the point lies in this Bame 

 plane, for it meets it once at the point and once on the simple director; 

 i. e. any plane through the simple director meets the scroll in this 

 director ami in three generators that intersect in the point where the 

 plane meets T. 



'1. Proof of Theorem I. — Pass a plane through 7'; it meet- the 

 .-•roll in 7' ami one generator and meets C 1 » in a points. Now if we 

 revolve the plane about T it will cut out, in succession, each genera- 

 tor of the scroll, and since the plane always met I in the same num- 

 ber of point-, say t points, on T, it inert- c > in the Bame number of 

 points, say « point- on each generator, where t -| a "• Since three 



_ " _ ~ " 



generators lie in a plane a < - and t > — . 



:;. Plane Ourves.- -A plane that dor- not pass through any line on 

 the scroll, i. e. an arbitrary plain-, meets the bctoII in a plane quartic 

 curve having a triple point on '/'. and Bince an arbitrary generator m< 

 the plain- once, ever} plane quartic is a I . 



A plane through one and only one generator cuts oul a plain- cubic 

 having a double point on '/'. through which the generator passes, mak 



tyley calls a line on a surface when the tangent plane to tin su 



i- different si each point along the lin 



