510 PROCEEDINGS OF THE AMERICAN ACADEMY. 



joins this (?« — 2)-tuple point to any point of Cm never contains more 

 than one other point of Cm ; nevertheless during its revohition around 

 the line, it contains successively all the other points of the curve. If we 

 take an ordinary point of Cm as vertex and construct a cone with Cm as 

 base, this cone will be of the {m — l)st order. This cone contains the 

 line joining the {in — 2)-tuple point to the vertex as an {m — 2)-tuple 

 edge, and can therefore have no double edges. The curve can therefore 

 liave no apparent double points when viewed from an ordinary point of 

 the curve. We therefore have h — m — 2;* thus a twisted curve of 

 order m that has an (m — 2)-tuple point has m — 2 apparent double 

 points. 



2. Let Cm be a twisted curve of order m having an (m — 2) -tuple 

 point at A (where 4 ^ m). This curve will lie on a cone of order two, 

 say Koi whose vertex is A ; for any plane through the point A can con- 

 tain only two other points of Cm, the lines joining these points to A 

 being generators of a quadric cone. Not more than two tangents to C,« 

 at A can lie in one plane, for the plane would then meet the curve in 

 more than m points, which is only possible if the plane contains the 

 curve. Every unicursal curve of order m with an actual (m — 2)-tuple 

 point thus lies on a quadric cone. We can show that Cm can be cut out 

 of this cone K2 by a monoid. Clearly Cm is the complete or partial 

 intersection of K^ with some surface of order /x, say S^, having a ^:-tuple 

 point at A. Now ??« — 2 ^ 2 k, since 2 ^ is the multiplicity of A on the 

 complete intersection of /S^, and iu It is evident that k is the order of 

 the cone that contains the 7n — 2 tangents to Cm at A that lie on Z^, 

 provided m - 2 ^ ^ k (k + 3) - ^ (k - 2) {k -\- I) - 1, where 2 ^ k, 

 i. e. provided m — 2 ^ 2 k. where 2 ^ ^. We can thus always take k = 



— - — or — - — , according as m is even or odd. The surface S^ with a 



^-luple point at A is determined by -^ (ju + 1) (;< + 2) (jm + 3) — ^ ^ 

 (^- + 1) (^ + 2) — 1 points. In order not to have S^ break up into K2 

 and a component of order ;u — 2 having a (^ — 2)-tuple point at A, we 

 must take ^ (ju — 1) |u (// + 1) — ^ (k — 2) (k — 1) k of these points 

 off K^. There are [x (^ + 2) — k' points remaining that can be taken 

 arbitrarily, but in such a way that S^ shall contain Cm- S^ meets Cm 

 in k (m — 2) points at A, and will therefore contain (7„, entirely if it 

 contains m ft — (m — 2) k -\- 1 additional points of Cm- Sfj. can therefore 

 be determined so as to contain Cm if 



* Salmon's Geometry of Three Dimensions (1882), No 330, Example 2. 



