WILLIAMS. — GEOMETRY ON RULED QUARTIC SURFACES. 37 



plane, is tangent to the generator in its plane where it meets the two- 

 fold director, but regarded as lying on the scroll, it does not meet 

 the generator there, for they lie on different sheets, and the generator 

 meets the cubic in one point only, where the plane is tangent to the 

 scroll — the formula giving (1 , 3 X ) = 1. The system of conies is the 

 same as on the scroll just considered. The two-fold director may be re- 

 garded as a 4 2 , since it has two lines on each of the two sheets through 

 it and is met twice by each generator. A plane quartic has a branch 

 on each sheet, and therefore meets the two-fold director four times, 

 (-!•!, 4 2 ) = 8 + 4 — 8 = 4, while a plane cubic has a branch on one sheet 

 only, and therefore meets this director twice, (3 X , 4 2 ) = 6 + 4 — 8 = 2. 



V. Quartic Scroll, with two Docble Linear Directors and 

 without a Double Generator, S (1», 1 2 , 4). (Caylet's First 

 Species, S (1 2 , 1 2 , 4).) 



1. We shall call the double directors D and 1/ . They do not inter- 

 sect, and if we pass a plane through either it cuts out two generators. 

 The scroll is similar in many respects to the Quartic Scroll S (1 2 , 1 2 , 2) 

 already considered, and what was said there in regard to gauch-polygons 

 applies equally well here. But the scroll now under consideration has 

 no double generator, the plane quartic directing curve having only two 

 double points, one on each double director, and this is the only quartic 

 scroll on which the multiple curve is of order less than three. 



2. Proof of Theorem I. — A plane through D outs out two genera- 

 tors that meet in a point where the plane meets D', and if we revolve 

 the plane about D, it will cut out, in succession, all the generators of the 

 scroll, two at a time. The plane meets C [a) in a points, of which a defi- 

 nite number, say 8 points, lie on D, and therefore there are a — 8 points 

 of C (a) on the two generators in the plane, taken together ; for any 

 given curve C'"\ the number a — 8 is a constant non-negative integer, 

 say k. Let x and y be the number of points of O a \ respectively, on the 

 two generators lying in a plane through D ; then as the plane revolves 

 about D, we always have x + y — &> and since x, y, and k are all non- 

 negative integers and k is constant, there is only a finite number of 

 values of x and of y that will satisfy this relation. Let us, for the 

 moment, designate any generator by the number of points of C" 5 on it, 

 i. e. the generator x has x points of O in) on it, etc. To any value of x, 

 say (j, there corresponds a certain value ofy, say g', such that g + g 1 =k; 

 if then a plane through D cuts out a generator g, it also cuts out a gen- 



