48 PROCEEDINGS OP THE AMERICAN ACADEMY. 



2. Proof of TJieorem 1. — By passing a quadric through E, which is 

 the cuspidal edge of the developable, and any chosen generator, Theo- 

 rem I is proved in exactly the same way as it is for the Quartic Scroll 

 S{^^, 1), p. 44, and if a be the number of points of C'(") on each gen- 

 erator, E has 2 (a — a) points of «„ on it. Since E is cut out by a 

 quadric through two generators, formula (1) holds for it (Theorem III). 

 Being a double twisted cubic met twice by each generator, E is a G^, and 

 formula (1) gives, for the number of intersections of E with a^, 



(62 ,«„)== 6 a + 2 a — 8 a = 2 (a — a). 



3. Plane Curves. — A " plane of the system," i. e. an osculating plane 

 of E, cuts out two consecutive generators and a proper conic* This 

 conic, which is tangent to E,1[ can never break up, for there are no lines 

 on the surface except the generators, and a plane cannot contain more 

 than two generators (which must, moreover, be consecutive), since it 

 cannot meet E in more than three points. Each conic is a 2i , and there 

 is a conic through every point of E. 



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

 a 3i , that has a cusp at the point not on the generator, where the plane 

 meets E, and has the generator for its real inflectional tangent at the 

 point where the generator meets E; for, since the two consecutive points 

 in which the generator meets E must be double points of the section of 

 a plane through the generatoi", the plane cubic must pass through each 

 of these points, i. e. it meets the generator in at least two points there ; 

 but the generator at this point crosses from one sheet to the other, say 

 from the upper to the lower, while the plane cubic crosses from the lower 

 to the upper ; therefore they cross each other and intersect, in an odd 

 number of points, and consequently they intersect in three points, i. e. 

 the generator is the real inflectional tangent. 



Every plane quartic has three cusps, one at each of the three points 

 where its plane meets E, and since it is the complete intersection of its 

 plane with the developable, formula (1) holds for it (Theorem II). 



Every conic is cut out by a plane through two generators and every 

 plane cubic is cut out by a plane through one generator, and therefore 

 formula (1) holds for every conic and for every plane cubic (Theorem 

 III). Formula (1) holds, therefore, for every plane curve on the 

 developable. 



* Salmon's Geom. of Three Dimensions, § 339. 

 t V. Jamet, C. R., 1885, Vol. 100, pp. 1332-35. 



