1270 
Also the 27 common bisecants of a’, 8°, y* taken two by two 
are singular lines of the /*. A common chord of a’, 6° bears oo’ 
pairs of points determined on it by the pencil (c’). 
§ 11. We now consider the locus 2 of the points P’ associated 
to the points P of a line /. To the points common to / and each of 
the surfaces A‘, G? correspond respectively 16 points of a? and 12 
points of g. Any surface a*_contains these 28 points Pand moreover 
the two triplets corresponding to the points common to a? and 1. 
So the locus À is a curve of order 17. 
As 7 contains eight coincidencies P=’ it is an eightfold secant 
of the curve 417; so any plane p through 7 contains 9 points P’ 
associated to points of /. So, the pairs of associated points lying in 
a plane generate a curve of order nine. 
The curve (G)° corresponding to the trace G of g meets p in 
four points; so G is a fourfold point of the curve ¢*. In an analo- 
gous way the nine traces Aj, be, Cx of the base curves are double 
points of p°. 
The intersection d® of g and the surface of coincidencies has a 
fivefold point in G. So gy’ and d° intersect each other in 9 Xx 8 — 
—-4 Xx 5—9 x 2 = 34 points differing from the traces of the bases. 
To these points belong the points of contact of the curves, corre- 
sponding to coincidencies of the /* the bearing lines of which are 
contained in gp. 
In order to determine their number we consider the three pencils 
of conics common to p and (a’), (h®), (c°). The polar curves of these 
pencils with respect to a point P deseribing a line / generate three 
projective pencils (a?), (0°), (c°). The first and the second generate a 
curve c° with G as node and passing through the three base points 
A, of «@ and the double points of the three pairs of lines. The curve 
6° generated by the pencils (a*) and (c?) also contains these points, 
So 6° and c° admit 25 — 4 — 3 — 8 —= 15 points of contact of three 
corresponding conics forming therefore coincidencies of the /* with 
a bearing line lying in gp. 
So g* and d* have four coincidencies in common the bearing lines 
of which intersect the plane ¢. 
