WILLIAMS. — GEOMETRY ON RULED QUARTIC SURFACES. 27 



a triple point ou the complete intersection ; the generator meets the 

 cuhic again where the plane is tangent to the scroll. Since an arbitrary 

 generator meets the plane once and does not meet the generator lying in 

 that plane, it meets the plane cubic once, and therefore every plane cubic 

 is a 3!. If a plane cuts out a proper conic, it must also cut out another 

 conic, which must be an improper conic consisting of two lines through a 

 point of the proper conic, since the section by the plane must have a triple 

 point on T; but the only lines on the scroll that pass through a point of 

 r l\ besides T itself, are generators, and we have seen that a plane through 

 two generators also cuts out a third generator and the simple director ; 

 therefore, there are no conies on this scroll. The triple director T is 

 met once by each generator, aud is, therefore, a 3^ The simple director 

 is met once by each generator, and is, therefore, a \ x . 



Each of these plane curves is either the complete intersection of the 

 scroll and a plane, or else the residual intersection consists entirely of 

 generators, and therefore by Theorems II and III, formula (1) holds 

 for every plane curve on the scroll. 



A plane quartic has a branch on each sheet where it crosses T, and 

 therefore meets T three times, as the formula shows, 



(4, , 30 = 4 + 3 - 4 = 3. 



A plane cubic has a branch on each of two sheets where it crosses T, and 

 therefore meets T twice, (3 t , 3j) = 3 + 3 — 4 = 2. The simple di- 

 rector does not meet 2] (1 1} 3j) = 1 -f- 3 — 4 = 0. The simple direc- 

 tor meets a plane once, and therefore meets a plane quartic once, 

 Oij 4 X ) = 1, but since it meets each generator once, it cannot meet a 

 plane cubic, (1 1 , 3 X ) = 0. Two plane cubics intersect in two points, 

 on the line of intersection of their planes, (3 X , 3 X ) = 3 + 3 — - 4 = 2, 

 the other two points, where this line meets the scroll, being the two 

 points where each cubic is met by the generator that lies in the plane of 

 the other. A plane cubic and a plane quartic intersect in three points 

 on the line common to their planes, (3 t , 4 X ) = 3 -f- 4 — 4 = 3, the 

 fourth point where this line meets the scroll being the point where the 

 generator in the plane of the cubic meets the plane quartic. Two plane 

 quartics meet in four points on the line of intersection of their planes, 

 (4 1 ,4 1 )=4+ 4-4 = 4. 



4. Iwisted Cubic 3 V — We saw that when a is odd we can take 



a -f 1 

 v = — - — ; so, for the twisted cubic, v = 2, and by formula (2), p. 23, 



wc have two points at our disposal in the determination of this quadric, 



