474 PROCEEDINGS OF THE AMERICAN ACADEMY. 



double points and points of higher multiplicity than two have no effect 

 on the number of apparent double points. Halphen* gives as the 

 lower limit of the number of apparent double points of a curve of order 



m the greatest integer contained in I — - — j ; the curve then being the 



intersection of a quadric cone and a surface of order — or — - — , accord- 



^ 2 2' 



ing as m is even or odd. According to this, the lower limit of the num- 

 ber of apparent double points of a quintic is four; and Salmon f gives 

 four quintics having four or more apparent double points. Genty, $ in 

 his consideration of the maximum number of actual double points of a 

 twisted curve, makes use of the lower limit of the number of apparent 

 double points as given by Halphen and applies it to all curves. Picquet, 

 in a note to Genty's paper, states that Halphen's lower limit holds 

 "pour une courbe gauche quelcouque." Dr. Williams, § however, gives 

 a quintic curve that has a triple point and only three apparent double 

 points ; thus one less than the number given by Halphen's formula. This 

 diminution is due to the fact that the curve has a triple point at which 

 the tangents do not lie in one plane. 



It is the object of this paper to consider curves that have multiple 

 points of different kinds and to investigate the effect of these different 

 kinds of points on the other singularities of the curves ; to consider the 

 curves first as the intersection, partial or complete, of any two surfaces 

 whatever that may contain them, and then as the intersection, partial or 

 complete, of a cone and a monoid; to consider the number of actual and 

 apparent double points to which a multiple point may be equivalent ; and 

 finally to consider in detail all quintic, sextic, and septimic curves that 

 have multiple points. In the consideration of the curve as the inter- 

 section of a cone and a monoid, we are led to the consideration of the 

 different kinds of lines possible on a monoid and the effect on the curve 

 of intersection when these lines are common to the cone and the monoid. 

 In the consideration of the composition of multiple points we shall only 

 consider points of multiplicity not greater than seven. 



* Loc. cit. p. 381. 

 t Loc. cit. p. 318. 

 t Loc. cit. p. 151. 



§ Geometry on Ruled Quartic Surfaces. Proceedings of the American Acad- 

 emy of Arts and Sciences, Vol. XXXVI. No. 3, July, 1900, p. 60. 



