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in four points associated to A; so ®*® passes four times through 
the base curves af, s*, y’. This is in accordance with the fact, that 
each trace of a base curve is threefold on p'* and onefold on d*. 
The curve d* contains 18 coincidencies the bearing lines of which 
lie in the plane, for the curve J®° ($ 4) corresponding to a line / 
of p meets / eight times. These 18 coincidencies lie on '°; so gy” 
and d° touch one another in 18 points. Moreover they have 36 
points in common in the 12 traces of the base curves; each of the 
remaining 48 common points belongs as coincidence to a group of 
the /* containing still one more point of gy’. 
§ 7. The plane p contains a finite number of associated triplets. 
As these triplets have to lie on p'* we determine the order of the 
locus of the sextuples of points P’’ associated to the pairs P,P’ of p'°. 
The surface A*’ passes eight times through §*, y* and one time 
through «ef. As p'° has threefold points in the 12 traces of the base 
curves it meets A? elsewhere in 15<32—4«3 — 2K4x«3xK8 = 276 
points forming 138 pairs P, P’ corresponding to 138 points P’’ of 
a“. A surface a? cuts y’* in the four threefold points A and in 9 
pairs P, P’ more, each pair of which determines six points P’’ on 
a. So the locus under discussion has 188 + 6 x 9 = 192 points 
with a? in common and is therefore a curve °°. Of its points of 
intersection with p a number of 48 lie in the points common to g'* 
and d° indicated above. Evidently the remaining 48 traces of p°° are 
formed by 16 triplets of the Z°. So any plane contains sixteen triplets 
of associated points. 
§ 8. If the bases of the pencils (a’), (0°), (c?) have the line g in 
common, three surfaces a’, ?,c? intersect each other in four asso- 
ciated points; so we then get an involution /* of associated points. 
Any point A of the curve a’ completing g to the base of (a*) 
belongs to oo' quadruples. These quadruples lie on the twisted cubic 
(A)? common to the surfaces 6°, c? passing through A and they are 
determined on (4)? by the pencil (a). 
In the same way any point ZB of the base curve p* and any 
point C of the base curve y° belongs to oo' quadruples. 
We determine the order of the locus A of the curves (A)*. By 
means of the points A the surfaces of (5°) and (c°) are arranged in 
a correspondence (4,4), any surface 4’ or c° containing four points 
A; so the surface A is of order 16. 
In any plane through g the pencils (4°), (c°) determine two pencils 
