15 
§ 5. Each straight line Bj Bi is evidently singular, for it bears oo! 
pairs of points that are harmonically separated by a’. 
A straight line would also be singular if the involution in which 
it is eut by (57), coincided with the involution of the pairs of points 
that are harmonically separated by a’. And this will be the case 
when this straight line is touched in its two points of intersection 
with a? by conics 6’. 
Now the straight lines ¢ that touch 6? at its points of intersection 
with a?, envelop a curve of the class sav. For the points of contact 
of the tangents out of any point to the conics 6’ lie on a cubic 
and this meets a? in six points, each of which defines a straight 
line ¢. This envelope is rational; it has therefore ten double tangents; 
to them belong evidently the six straight lines bz by. 
Hence there are, besides these, four more singular straight lines, sp. 
The straight line sz is transformed through (2,2) into the system 
of sz, and a nodal cubic that has its double point in the pole of sz. 
The straight line Bz B; is transformed into the system of bj Bi, 
Br B by and 6. 
§ 6. The points P, and P, associated to Pin the (2,2), correspond 
to each other in another (2,2), which may be called the derivative 
of the former. This (2,2)* is likewise of the jirst class; for on a 
straight line p there lies only the pair in which p cuts the conic 
6? passing through the pole P of p. 
Also this (2,2)* has singular points. in Bz; for if P describes the 
polar line bz, P, remains in B, and P, describes the above mentioned 
rational curve 37. 
The curves 0,‘ and 9,‘ corresponding in the (2,2) to the straight 
lines 7, and r,, have ($ 1) 10 points P in common for which P, 
lies on r,, P, on r,. Hence P, describes a curve o'° when P, 
describes the straight line 7,. This o'° has quadruple points in By, 
for r, cuts the curve 3,* in four points P,. 
Each branch point of the (2,2) is at the same time a branch point 
of the (2,2)*; accordingly they have also the same branch curve (V )°. 
The coincidences of the (2,2)* are the double points of the (2,2); 
the curve of coincidence is therefore the above mentioned d*, which 
passes three times through B, twice through the double points of 
the pairs of lines. We find the points of intersection of 7 with g@*° in 
the eight points which r has in common with d* and in the pair 
orpoints, 2, P oom 1e 
The four singular straight lines ($ 1), of the (2,2)* are found in 
the straight lines dy. 
