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wc can puss to tlie analysis of liie principal curve. It consists of 

 tlie following parts: 



a. The conic K-d in plane t>\ , locus of the points corresponding 

 to pencils of screws having a ray in common with the pencil (^, a) 

 and whose planes ar^ parallel to the nodal line </. 



h. The conic lOa in plane I'l , locus of the points corresponding 

 to pencils of screws having in conmion a ray with (.1,«) and con- 

 sisting of parallel rays. 



c. The line li^DuD,,, intersecting K~,i and K"u , locus of 

 the points corresponding to pencils of screws whose vertices lie on 

 a' and whose planes pass through «. 



Both conies have two imaginary points in common; the planes 

 ö^ and vi are the loci of the points cori'csponding to screws pa- 

 rallel to d and to those in the plane at inhnity. They have not 

 been indicated here. 



In the figure have further been constructed the vertices 7',, and 7)n of 

 the two quadratic cones, determined by A'-j and K\ . The line 

 connecting 7'^ , 7^ meets ^\ and r^ in J\Jd and 7l/„ . 



G. Now the forms of JE^ Curresponding to the cones and parabolas 

 of ^ can be found. It is evident that we shall get curves in ^i- 

 Let us take a vertex P and construct a cone of screws F^. The 

 construction gives rise to the following remarks: 



a. Let us imagine through (A, a) a zero system, formed by two 

 reciprocal systems of points and planes with the property that any 

 point lies in its corresponding polar plane ("Nullsystem" of MüEBiüs), 

 with a linear complex o situated in it; the polar plane of P inter- 

 secting cone P^ in two generatrices, the corresponding plane '^i inter- 

 sects in two points the curve Pj'-^ cori'csponding to P^; so Pj^ is 

 a conic. 



b. One screw of 7^^ intersects a and o'; so Pj- meets /i in one point. 



c. One screw of 7^^ is parallel to d-^ so P^' intersects Si in one 

 point not situated on A'^,; . 



d. It is very important to determine how many screws of P' 

 belono' to the pencils of parallel screws having one screw in common 

 with {A, a). Let us bring a plane parallel to a through 7^; two 

 screws of P^, m and »', are parallel to the screws m' and it' of 

 pencil {A, «); from this we conclude that m and m' are perpendi- 

 cular to the same generatrix of C^ ; so they belong to the same 

 pencil of parallel screws, viz. to a pencil having with {A, a) one 

 screw in common. The same can be said of « and »i'; so 7^1^ inter- 

 sects the conic A'«- in two points. 



