1265 
M is sevenfold on the locus of the pairs P, P’ collinear with 1, 
and this locus is a twisted curve (/)'* passing seven times through J/. 
The curve (P)'* is common to the surfaces (Q)’ and (R)’, inter- 
secting each other moreover in the curve of order 15 common to 
-(Q)* and the polar surface M*; so the residual intersection consists 
of 11 lines. The lines are singular chords of the bilinear congruence’) 
of the curves of = (b’,c’), i.e. any of these lines contains o' pairs 
(Q, Q’); these lines are not singular for /*, as these quadratic invo- 
lutions have only one pair in common. 
Amongst these 11 lines we find two chords of 8* and two chords 
of y*. So the complex T° contains three congruences (2, 6) and three 
congruences (7,3) the rays of which are singular chords of a bilinear 
congruence (of). 
There are 120 lines g each of which contains oc! pairs of the 7°, 
i.e. the common biseeants of the base curves a’, 6‘, y‘ taken two 
by two. A common bisecant of af and 8* forms, in combination 
with a twisted cubic, the intersection of an a? and a 0°; evidently 
any pair of the involution determined on it by the pencil (c’) is a 
pair of Z*. So this involation admits 120 singular chords. 
The curve (P)’’ cuts each of the base curves in 20 points, as 
the surface (Q)° corresponding to M has 20 points Q in common 
with a‘; the surface a’ containing the corresponding point Q’ also 
contains Q, ie. Q, Q’ is a pair of the /°. 
The three polar surfaces of J/ with respect to the pencils (a®, 
(57), (c?) intersect each other in M and 26 points more; in any of 
these points A the line WR is touched by three surfaces a’, b?, c?. 
So A is a coincidence P= P’ of the /*, the bearing line passing 
through M, So the twisted curve (P)'* admits the particularity that 
26 of its tangents concur in the sevenfold point M. 
§ 4. If M describes a plane 4, the three polar surfaces generate 
three projective nets. The locus of the points of intersection consists 
of the plane 4 and a surface A containing all the coincidencies of 
the J’. 
We deduce from 
A AAD 
EP =| 
OENE 
that this surface is of order eight. *) 
1) loe cit. 
*) This result is in accordance wilh a theorem of Mr. G. A@uGita (Sulla super- 
