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points P; as II’ intersects the twisted cubic 4° described by P when R 
moves along / in nine points, the correspondence (#,S) is of order nine. 
A point of coimeidence of A and S ean only present itself when » 
and s coincide, i.e. on a common chord 6. On any of the ten 5 the 
pairs (P,R) and (PS) generate two involutions, of which A,, H, 
may represent the common pair. By assuming A in Hy, we find 
fH, for P and FH; for S; so H, and H, are points of coincidence 
of (RS). So this correspondence admits twenty coincidencies lying in 
pairs on the fen common chords 5. 
As a point 2, of y® corresponds to each point P of the tangent 
r, of Fi. R, corresponds to each point S of the twisted cubic c,° 
into which r, passes by the transformation (P,S); evidently 5,° has 
four points in common with o°. 
Consequently the curves g' and o' are singular curves of the 
correspondence (R‚S). 
If B describes the tangent r, of 9°, P remains in the point of 
contact of 7,; so the point S* conjugated to P is singular and corre- 
sponds to all the points of 7,. Evidently the locus of S* is the rational - 
twisted 6° into which «’* passes by the transformation (P,3S). 
So the correspondence (2,S) admits too singular twisted curves of 
> 
order nine, 6° and ©’. 
As the developable with g° as cuspidal curve cuts 6° in 12 points 
a? and o° have twelve points in common; likewise g° rests in 12 
points on «°. 
§ 3. We now consider the*“lines p= RS. If P describes the 
line /, p generates a scroll of order six; for we found above that 
the plane a == P,, passes through / in six positions (§ 1). 
The line p generates a complex. We determine the number of 
lines p belonging to a pencil with veriex 4 and plane A. 
If R describes a ray / of pencil (L,4), S generates a curve 
intersecting À in nine points ($ 2); we conjugate to / the nine lines 
/’ connecting these points with JZ. In this manner we get in the 
pencil a correspondence (9,9) each coincidence of which furnishes a 
line p connecting two points Rand S corresponding to each other. So : 
The complex (p) is of order eighteen. 
Evidently the 20 points H are principal points of the complex ; 
each complex cone passes through these 20 points. 
§ 4. Any point &, of 9’ issingular, for it bears the lines p connecting 
it with the points S of the corresponding curve o,’ ($ 2) and so its 
