21 PROCEEDINGS OF THE AMERICAN ACADEMY. 



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

 residua] consist of curves of orders less than a. 



[n determining the number of points at oar disposal, given by formula (2), 

 we have said nothing about multiple points on C"'\ but have supposed 

 that the points of <SW that had to be taken on were ordinary points 

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

 multiple points on C"K 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 S") pass through a v + \ ordinary points of C'" in 

 order to make $") contain C (r " ; but if C w has a double point, any surface 

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

 if we make SM pass through this double point (which counts for only 

 one point in the determination of S 1 -'')), we have to make it pass through 

 only a v — 1 other ordinary points of C"' in order to make it contain 

 C"\ since C {,,) will then intersect <Sv*) in a v — 1 + 2 = a v + 1 points. 

 Consequently, when C {a) has a double point, only a v of the points neces- 

 sary to determine *SW 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 Sfc ' taken 

 on C' ( "' in deduciug 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' 1 " 1 has an ///-tuple 

 point, a surface S^ through that point meets C io) m times there, and we 

 need only make S ''» pass through a v— (in — 1) other ordinary points 

 in order that it shall contain C { " ; and, consequently, when C ,a) has an 

 ///-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 v + 1 of the points of intersection of 



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

 looked, viz., if a curve has an m-tuple point /' and the //< tangente to the curve at 

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

 Pm an ordinary point, and any line /- through /'. th.it .lues net lie in the tangent 

 plane, meets this BUrface only once at /'. but there an //, br I the curve 



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

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

 point on any surface that contains the curve, the multiplicity I of /' on any such 

 surface being at least equal to the onler of the cone of lowest order that cai 



ed through the //, tangents, ami this lowest order i- always less than m. Then 

 any line /. that does not lie on the tangent cone to the surface .-it /' meets the r-ur- 



timea at P, while ai fthecot ts the surfa I ; 



