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straight lines, which form a plane pencil that has the intersection 

 of / and c for vertex. 



If ;. changes continuously, out of a system S of oo' straight lines 

 through the homographic representation described above, co^ new 

 systems are derived forming a coherent set, which contains *S' (for 

 A = 1) and of which we shall especially consider S' , the system 

 arising from S through the transformation belonging to X = 0. The 

 number of straight lines which a system of this set has in common 

 with a system of oo' straight lines not belonging to the set, is apparently 

 independent of ).. In order therefore to know how many straight 

 lines aS has in common with another system S^ of oo' straigh lines, 

 we may equally well investigate the same for S'. 



Now any straight line of S is transformed into the line c, which 

 may always be chosen outside S^. If, however, -S contains ^ straight 

 lines / passing through an arbitrary point, so that k is the class 

 of the curve enveloped by the lines /, the k straight lines of S 

 through are transformed into as many plane pencils of straight 

 lines /'. S^ contains k^ lines of each of these plane pencils, if the 

 straight lines of aS'^ envelop a curve of the class k\ From this we 

 conclude that *S'' and S\ hence also S and S\ have kk^ straight 

 lines in common. 



^ 3. In order to apply the same method to the straight lines of 

 space, we assume a point C and a plane y, and we make use of 

 the homographic representation arising if to each point P we associate 

 the point P' that forms with C, the point of intersection C' of CP 

 with 7, and P an anharmonic ratio =^; in this representation there 

 corresponds to any straight line / another straight line /' cutting / 

 on y. If again we take the case ). = 0, to any straight line / a 

 straight line /' of y is associated, the intersection of the plane (C,/) 

 with y, unless / passes through C in which case there are oo' 

 associated lines /', which form a sheaf of rays that has the point 

 of intersection of / and y for vertex. 



In this way a ruled surface R is represented in a system R' of 

 00^ rays /' of y. These envelop a curve of the class q, if q is the 

 order of R. For through a point P of y there pass those straight 

 lines /' that are the images of the straight lines / of R cutting CP. 

 If now we consider a complex /{ of the order x, this has in y 

 oo' rays enveloping a curve of the class jc, so that K has xq rays 

 in comuion with R' . 



A line complex of the order z has therefore xq lines in common 

 luith a ruled surface of the order q. 



