( 482 ) 



brought through a point .1, of ;'i and through }\ , cnts out of *P'' a 

 curve I" of order n and class /? (?z — 1), from which ensues that 

 the tivo directrices }\ , r.^ are n [n — 1) fold lines of the scroll ii 

 under examination. 



A plane through ?'j contains the n {n — 1) fold line r^ , likewise 

 the 71 {n — 1) single generatrices through the point of intersection of 

 that plane with r^ -. so ^i is a surface of order 2n {?i — 1). 



Let >Si be a point of intersection of Tj and 0. The plane S^r^ now cuts 

 *P according to a k", containing the point .S', itself, from which ensues 

 that two of the Ji {7i — 1) generatrices of ^ through /Si coincide with 

 the tangent in S^^ to X,"; throtu/h each of the n points S^ passes 

 therefore a torsal line of ii, and the tangential plane belonging to it, 

 irkich for convenience sake loe shall call "torsal plane", is evidently 

 the plane S^r^ . The same holds of course for r, . 



There are however more cuspidal points on r^ . If namelv we 

 imagine a tangential plane through r^ to ♦?>, then it will intersect <l* 

 in a /(•'' with a notie in the point of contact; the line connecting this 

 point of contact with the point of intersection C\ of the indicated 

 tangential plane and ;\ connt'^ for two coinciding generatrices of 5i 

 through Ci and is thus likewise a torsal line; so the points C^ 

 are also cuspidal points of i2. Their number is equal to the class 

 of 0, thus n {n — 1)% and I lie corresponding torsal planes are the 

 planes C^r.^ . The same holds of course for r, . 



Other cuspidal points on ;■, or r, are not possible. For, if for a point 

 A^ of r, two tangents to the curve I'" lying in the plane A^r^ are to 

 coincide, then this is only possible either in one of the manners 

 described just now or because an inflectional tangent or a double 

 tangent of t' passes through A^ . These last cases appear in reality 

 (comp. §§ 4, 6), however, they evidently do not lead to torsal lines, 

 but to cuspidal edges and nodal generatrices. The complete number 

 of cuspidal points on r^ (or r^,) amounts therefore to 

 n -|- n i^n—iy = n {n-—%i-\-2). 



§ 3. As each generatrix of i2 is a tangent of ^P the scroll i2 and 

 the surface *P will touch each other along a certain curve, whilst 

 both surfaces will possess in general a proper curve of intersection 

 besides; for, of the n points of intersection of a generatrix of ^ 

 with *P only two (coinciding ones) belong to the curve of contact, 

 the remaining n — 2 to the curve of intersection. 



The order of the curve of contact we can find in the following 

 way. A plane through r, and a point A^ of r^ intersects *P in a 

 curve h, and the points of contact of the tangents drawn out of 



