Mathematics. — "A New Method for the Solution of the Problem 

 of the Characteristics in the Enumerative Geometry." By 

 G. ScHAAKE. (Communicated by Prof. Hkndrik db: Vriks.) 



(Communicated at the meeting of April 2!), 1922). 



^ 1. In this paper a general method will be set forth for the 

 determination of the expressions through which the problem of the 

 characteristics in the enumerative geometry is solved. These are the 

 expressions indicating how many individuals two algebraical systems 

 resp. of cx)/' en oo"-/' figures, depending on n parameters, have in 

 common. 



The method in question will be best explained by application to 

 a special example. We shall therefore by the aid of it solve the 

 problem of the characteristics for the straight line in a space of an 

 arbitrary number of dimensions. ^) 



§ 2. We shall tirst confine ourselves to the straight lines of a 

 plane V. In V we assume a point 6' and a straight line c by the 

 aid of which we represent tlie plane homographically on itself. 

 With a view to this we associate to a point P of V the point P' 

 of the straight line CP that together with 6', the point of inter- 

 section C' of CP and c, and P' forms an an harmonic ratio that 

 is equal to a constant number A. Through this transformation a 

 straight line / of V is transformed into a straight line /' cutting 

 / on c. 



Especially we consider the transformation for which A = 0. In 

 this case for an arbitrary point P the distance C'P'=0, so that 

 the point P' corresponding to a point P generally lies in the inter- 

 section of CP with c. If, however, P lies in 6', together with the 

 straight line CP also the distance C' P' becomes indefinite, so that 

 to the point C all the points of V are associated. 



P'or an arbitrary straight line / the coriesponding line /' coincides 

 with c. If, however, / passes through P, there are ao^ associated 



') Cf. for other applications Cap. VI of my academical dissertation which will 

 shortly appear, entitled: Afbeeldingen van figuren op de punten eerier lineaire 

 ruiwie, Groningen, P. Noordhoff, 1922. 



