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point S on (J, is out in (n + 1) points by an arbiirary straight 

 line /; consequently <J is an {?i -}- l)-fold curve on the sui-face A of 

 the curves y", wiiich are cut by /. As two surfaces A apart from 

 Ö can only have in common a number of y", which agrees with 

 the order of yi, we have foi- the determination of that order -v the relation 



w' = nx -f {7i + 1)' {n' + ?< + !); 



from which ensues .x = {n -\- l)^ 



The y^ restimj on a straight line I form a surface of order 

 [ji _[_ \y on ichich the 7", of ivhlch the plane passes through I, is an 

 n-fold curve ; the singular curve is (w -}- l)-fold. 



A is cut n{n-\-'\)" times hy an arbitrary y" of the congruence; 

 from this appears again tliat y" rests in n in -f- 1) i^oints on 0. 



Two arbitrary straight lines are cut by [n -J- I)"' curves of the 

 congruence. 



A plane (f^ passing throug[\ / intersects yi moreover along a curve, 

 which is apparently cut n {n — 1) times on / by the y", of which 

 the plane passes through /; in each of the remaining {n -{- ly — 1 — 

 71 (n — 1) = 'Sn points (/ is touched by a y". 



The curves y", lohich touch a given plane have their points of 

 contact on a curve of order 3/i, ivhich possesses {if-{-'n-^l) double points. 



Tlie last mentioned observation ensues from the fact that the 

 surface JS'«+", which has a node in a singular point aS, is cut by 

 f/j along a curve with node S ; (f is therefore touched in *S by two y". 



Tlie curve (/^" found just now is the locus of tlie coincidences of 

 the involution formed from collinear sets of 71 points in which (f is 

 cut by [y"]. 



3. The surface J belonging to an arbitrary straight line, not 

 lying in </), has apart from the {n" -\- n -\- i) points S S7i{n-{-iy — 

 2 (7i+l) {n'^n^l) = {n-i-l) (n'-\-7i—2) = (?i-[-2) {n'—l) points in 

 common with ip^'K 



'There are {71 -{- 2) {n" — 1) curves in [y], lohich touch a given 

 pla7ie, a7id at tlie same time cut a given .straight liiie. 



We can arrive at the last mentioned result in an other way yet. 



The surface ^"+', which contains the y", the planes of which 

 pass through a polar ray ƒ, is cut by a straight line /in {71 -\- 1l) 

 points ; so the planes of (?i + 1) curves y" pass through ƒ, which 

 curves rest on /. Consequently the planes of the y" lying on ^ envelop 

 a cone of class (n ~{-l). 



A plane ff> cuts ^"+' along a curve r/"+i, which passes through 

 the point of intersection of/, and sends {7i-\-l)n — 2 = {n-\-2) {7i — 1) 



