514 PROCEEDINGS OP THE AMERICAN ACADEMY. 



Substituting in formulae (4), we have 



Comparing (5) and (6), we have 



$i = h — mk + k(k+ 1 ) ; 



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



when viewed from a Avtuple 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 7;;-tuple point. Then the curve can have no more actual double or 



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



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



(„, _ k - 1) (m - k - 2) , ,, 



have more than double edges. I he 



2 s 



(m — k — 1) -tuple edge to the (m — k — l)-tuple 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 fc-tuple 



point. It has therefore just m k — k (k + 1) apparent double points 



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



apparent singularities. 



Let P n denote an rc-tuple point on the curve. A curve of order m 



can thus have a P m —k-\, a P^ and [mk — k (k + 1)] apparent double 



points, or a P m —i;, a -P/c—i, and \jn (k — 1) — k (k — 1)] apparent 



double points. We can therefore write symbolically 



P m -k + Pk-l = Pm-k-x + Pk + (m — 2 k) apparent double points* (I) 



[We obtained m k — k (k + 1) apparent double points by assuming the 

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

 P m -k—\, we get the same result, viz. : 



Pic, P m -k-h and m (in — k — I) — (m — k — 1) (m — k) 



apparent double points, i. e. 



P/c, P m -k-h an d mk — k (k + 1) apparent double points.] 



