445 



passing through P determine a net and have the base-points of this 

 net in common. 



P'oi- the more general /" the proposition of § 1 does not hold 

 good ; consequently the joining lines of associated points form a 

 complex of rays instead of a congruence of rays. 



The locas of tiie coincidences is now a surface of order 8; the 

 curve associated lo a straight line / is of order 23, the surface 

 associated to a plane V is also of order 23. The question arises 

 how the results obtained above are connected with the properties 

 of tliose more general 1'. 



§ 10. If the 3 pencils (</>^) lie in the same complex go' pencils 

 {A'^) may be introduced intersecting the three given pencils. If the *° 

 of the complex are represented by the points of a tridimensional space, 

 the {A'') are represented by the generatrices of the ruled surface 

 having tiie images of tiie given «f»" as directrices. 



For a point P on the base-curve X'' of a (J^) the three <ï>- from 

 the given pencils passing through /■* belong to (.P), consequently 

 they have /' in common. For siirh n point P the associated points 

 Q hi'coini' therefore indefinite, if we start for the definition of the [* 

 from the three pencils {<!>-) instead of directly from the complex. 



In order to find the locus of P, we observe that the 'ƒ>" of the 

 three pencils (*'} belonging to one and the same pencil {A^) are 

 projectively associated to each other, as immediately follows from 

 the representation mentioned. The base-curves ).* are consequently 

 sections of corresponding surfaces <I>- out of two projectively 

 associated pencils; their locus is {hereïore & surface of order four, il*. 



§ 11. If starting from the niore general /*, the given pencils <?'■' are 

 allowed to change in such a way that they come to lie in the same com- 

 plex, the occui'rence of ii^ will a[)parenfly cause various degenerations. 



As the points associated to a point /-* of i2^ are indefinite they 

 may also be considered as coinciding with P, and consequently the 

 surface A" of the coincidences of the general /" will degenerate into 

 A^ and ^i\ 



A straight line / intersects i2' in 4 points, intersects therefore 

 four /', the «j" associated in the general case to / degenerates 

 conse(piently into the y' found above and those four V. 



A plane V passing through / intersects i/^ in general in 15 points 

 outside /, of these 12 lie now on il\ which are associated by 3's 

 to 4 points of /. 



From the section of V with the associated surface 0" the section 



29* 



