1152 
As no two cotangential points can coincide on a not-degenerate o*, 
this 7° has no surface of coincidence. 
If P describes a straight line /, the associated points P’ describe 
a curve x. We easily find then that it has 16-fold points in the 
base-points Bz and is of order 72. 
If P describes a plane V, the points P will describe a surface Z, 
which is of order 72, and has 32-fold points in the points Bk, 
§ 10. By the results acquired here, we can also find the corre- 
sponding numbers for the bilinear congruence of twisted quartics ' 
which are formed by the intersections of the quadries of two given 
pencils (a?) and (b°). 
We first determine the locus of the points of inflevion of these 
curves. 
Let a° be a surface of the pencil (a). The @* lying on a’ pass 
all through the 8 intersections of a° with the base-curve p* of (6°). 
They consequently form a system entirely corresponding to that of 
the 9*, which in this case lay on one and the same ®*; the locus 
of their points of inflexion is therefore a curve of order 32, which 
has sertuple points in the intersections mentioned. 
So it is to be seen that the locus wanted has the base-curves 
a’ and g* as sertuple curves. A surface a’ intersects this locus now: 
1 along the o0* found just now; 2 along the sextuple curve a‘; 
the order of the locus is therefore 28. 
In the same way we find: 
The polar surface of a point P with regard to the congruence [94] 
has a‘ and #* as triple curves, and is of order 15. 
The satellite has «* and B as nonuple curves, and is of order 39. 
§ 141. If a point P describes the curve a', the tangential points 
of P will describe a surface t,. 
A surface a? intersects tz, besides along a*, in the locus of 
the tangential points of the points of a* lying on a‘. Now the 
curves 9* lying on a’ form a system entirely corresponding to the 
one which in the case of a net lie on one and the same @®; the 
locus mentioned is therefore the same as with a net [®*| that should 
have the 8 intersections of a? and 8 as base-points. Applying the 
results of § 8, we find, that this locus is a curve of order 4 x 37 
= 148, which has 36-fold points in the base-points mentioned. 
From this it ensues further that B* is a 36-fold curve of the surface tr. 
A surface 6? intersects rt besides along 8“, in the locus of the 
tangential points of the 8 intersections of 6? and a‘. According to 
