26 PROCEEDINGS OP THE AMERICAN ACADEMY. 



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

 S(^>n~,n) to denote a scroll generated by a line that meets a curve of 

 order tn twice and a curve of order ?i once, and S (m^) to denote a scroll 

 generated by a line that meets a curve of order m three times. In his 

 symbols for the quartic scrolls he has also used a subscript, in most cases, 

 to denote the order of multiplicity of the curve on the scroll ; but he has 

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

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

 form to his general rule for such symbols. 



II. Qdartic Sckoll, with A Triple Linear Director and a 

 Simple Linear Director, *S'(l3, 1, 4). (Cayley's Third Species, 

 '^(13,1,4).) 



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

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



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 same 

 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 and in three generators that intersect in the point where the 

 plane meets T. 



2. Proof of Theorem I. — Pass a plane through T; it meets the 

 scroll in T aud one generator and meets C") 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 meets C^") in the same num- 

 ber of points, say t points, on 2] it meets C(") in the same number of 

 points, say a points on each generator, where t -\- a ^ a. Since three 



... , — a , _ 2tt 



generators lie m a plane a <; - and t > -— . 



o o 



3. Plane Curves. — A plane that does not pass through any line on 

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

 curve having a triple point on T, and since an arbitrary generator meets 

 the plane once, every plane quartic is a 4i. 



A plane through one and only one generator cuts out a plane cubic 

 having a double point on 7\ through which the generator passes, making 



* Cajley culls a line scrolar on a surface when the tangent plane to the surface 

 is different at each point along the line. 



