514 PROCEEDINGS OP THE AMERICAN ACADEMY. 



Substituting in formulae (4), we have 



Comparing (5) and (6), we have 



^i = k — m k + k (k + I) ', 



that is, the number of apparent double points is less hy m k — k (k + 1} 



when viewed from a /c-tuple point on the curve than when viewed from 



an arbitrary point in space. 



Consider now a curve of order m that has an (m — k — l)-tuple point 



and a A-tuple point. Then the curve can have no more actual double or 



multiple points. It will in general lie on a cone of order tn — k that has 



the /c-tuple point as vertex and the curve as base. This cone cannot 



(m - 7v - 1 ) ( m - k- 2) 

 have more than double edges. ihe 



{m — k — l)-tuple edge to the {m — k — l)-tup]e point counts for just 

 this number. The cone can therefore have no double edges, that is the 

 curve can have no apparent double points when viewed from the /c-tuple 

 point. It has therefore just mk — k {k -\- 1) apparent double points 

 when viewed from an arbitrary point in space. It can have no more 

 apparent singularities. 



Let Pn denote an w-tuple point on the curve. A curve of order m 

 can thus have a Pm—k-i, a ■P^•• and [mk — k (k + 1)] apparent double 

 points, or a Pm-k, a Pt-i, and [m {k — \) — k {k — 1)J apparent 

 double points. We can therefore write symbolically 



Prn-k + Pk^\ = Pm-k-t + Pk + (m — 2 k) apparent double points. (I) 



[We obtained mfc — k (k + I) apparent double points by assuming the 

 vertex to be taken at Pk. If we interchange and take the vertex at 

 Pm~k—\, we get the same result, viz. : 



Pi., Pm-k-u and m (m — k — I) — (w — k — 1) {m — k) 

 apparent double points, i. e. 



Pa-, Pm-k-], and mk — k {k + 1) apparent double points.] 



