1266 
The coincidencies of the involutions T* lie on a surface &* passing 
through the base curves a*, B*, 7%. 
The surface A* also contains the three curves of order 14 con- 
taining the points of contact of surfaces of two of the pencils. 
The three polar surfaces generate three projective pencils if M 
describes a line /. These surfaces generate the line 7 and moreover 
a twisted curve d forming the locus of the coincidencies P= P’, 
the bearing lines of which rest on /. If the three pencils are 
indicated by 
Ar AMA VBB re 
the twisted curve under consideration can be deduced from 
| 
A B C 
= << Ti —0. 
A 3 C° | | 
x z Z | 
So the degree of this curve is 6? — 3? — 1 = 26.1) 
The line / bears 8 coincidencies, so it is an eightfold secant of d°*. 
§ 5. We now consider the locus of the points P’ associated to 
the points P of the line 7. The curve a‘ contains 32 points P’, as 
/ intersects A*? in 32 points. So any surface a? contains these 32 
points and moreover the two sets of seven points P’ associated to 
the two points common to a? and /. So the groups associated to 
the points of a line lie on a twisted curve of order 23, intersecting 
each of the three base curves in 32 points. In its points on A® the 
line / meets its curve 4**; so 1 eightfold secant of 22°. 
A plane y through / meets à° in 15 points not lying on 7; as 
these points are associated to 15 points P of J, the locus of the 
associated pairs lying in a plane is a curve of order 15. 
This curve, °°, has threefold points in the 12 traces of the curves 
a’, BY, y* on p. The curve (A*) corresponding to any of these traces 
meets y in three other points, each of which forms with A a pair 
Of ite: 15 
§ 6. The sets of seven points P’ associated to the points P of 
a plane p lie on a surface #** intersecting g according to the curve 
p'* containing the pairs P,P’ lying in @ and to the curve d° of 
the coincidencies lying in g. 
The curve (A)* corresponding to the point A of a‘ (§ 1) meets p 
ficie luogo di un punto in cui le superficie di tre fasci toccano una medesima 
retta, Rend. del Circolo Mat. di Palermo, t. XX, p. 305). 
1) AGuGLIA, |. c. p. 321. 
