24 PROCEEDINGS OF THE AMERICAN ACADEMY. 



greater by one than this lowest value, in order to be able to make the 

 residual consist of curves of orders less than a. 



In determining the number of points at our disposal, given by formula (2), 

 we have said nothing about multiple points on (7'"', but have supposed 

 that the points of /S^") that had to be taken on C*"^ were ordinary points 

 on this curve, and we shall always consider v chosen without regard to 

 multiple points on C"'. If a surface be made to pass through an ordi- 

 nary point of a curve it meets the curve once at that point, and therefore 

 we have to make aSI") pass through av-\-\ ordinary points of C""' in 

 order to make »S'(*') contain C'"' ; but if C^"' has a double point, any surface 

 through this double point will meet the curve twice there, and therefore, 

 if we make S''-") pass through this double point (which counts for only 

 one point in the determination of jS^"'), we have to make it pass through 

 only a V — I other ordinary points of C'"' in order to make it contain 

 C'"\ since f '"' will then intersect aS'") inav — \-\-2 = av-\-\ points. 

 Consequently, when C"' has a double point, only a v of the points neces- 

 sary to determine -S^") need be taken on C^"' if we take the double point 

 to be one of these : this is one less than the number of points of S^^) taken 

 on C'"' in deducing formula (2), above ; and therefore, when C'"' has a 

 double point, we shall have at our disposal one point more than the 

 number given by formula (2). In like manner, if C*"' has an m-tuple 

 point, a surface 6W through that point meets 6*'"' m times there, and we 

 need only make S'-"^ pass through a v ~ {m — I) other ordinary points 

 in order that it shall contain C'"' ; and, consequently, when 6'"" has an 

 m-tuple point, we shall have at our disposal m — 1 points more than the 

 number given by formula (2). 



In accordance with this principle, it is evident that, if v + 1 branches 

 of C'"-^ meet any line L* (i. e. if i/ + 1 of the points of intersection of 



* It is necessary to observe here a very important fact, whicli is often over- 

 looked, viz., if a curve has an m-tuple point P and the m tangents to the curve at 

 P all lie in the same plane, a surface on which the curve lies may have this point 

 Pas an ordinary point, and any line L through P, that does not lie in the tangent 

 plane, meets this surface only once at P; but there are m branches of the curve 

 that meet L at P and a plane through L meets the curve m times at P. In gen- 

 eral, if the m tangents at P do not all lie in the same plane, P will be a multiple 

 point on any surface that contains the curve, the multiplicity Ic of P on any such' 

 surface being at least equal to the order of the cone of lowest order that can be 

 passed through the m tangents, and this lowest order is always less than ?». Then 

 any line Pthat does not lie on the tangent cone to the surface at P meets the sur- 

 fa(;e only k times at P, while any edge of the cone meets the surface k + 1 times 

 there. 



