720 
coincidences” in the sense of ScHUBERT, i. e. as coinciding lines whose 
point of intersection as well as whose connecting plane is indefinite ; 
so they satisfy the delinition we have given above of torsal lines of 
the first kind. 
In the second place now however an s, can touch the curve &”; 
in this case the two coinciding rays s conjugated to the point of 
contact form a “single coincidence’, i.e. two coinciding rays whose 
point of intersection and whose connecting plane both remain de- 
finite; the point of intersection lies on /, the connecting plane however 
does not pass in general through /, for then it would be necessary that 
in the point where s, touches the curve £* at the same time also 
one of the cubic curves through the vertices were to touch that 
curve, which can of course in general not be the case; so we find 
torsal lines of the first kind. However, if there really were torsal lines 
of the third kind, then there would have to be among the points of 
contact of the rays s, with £’? also some where at the same 
time a cubic curve were to touch £'*; these particular points of 
contact would then give rise to the torsal lines of the third kind. 
The cubic conjugated to a plane 4 through / may have with 4% 
two coinciding points in common; in this case two generatrices 
lying in the same plane 24 coincide. This happens in the first 
place for those planes 2 whose conjugated cubic passes through one 
of the six nodal points of 4'?, and so we find again the nodal lines 
of 2°; this, however, also takes place if a cubic touches £**, and 
then we find a torsal line of the second kind; for the two rays s 
conjugated to the point of contact coincide whilst their connecting 
plane 4 remains definite. Their point of intersection les in general 
not on /, because the point of contact of £* with the cubic is in 
general not a point of contact of 4'* with a generatrix s, of the 
regulus; for those points however where that might be tbe case we 
would find torsal lines of the third kind. 
We calculate the complete number of points, where a line of the 
regulus has two coinciding points in common with £'*, with the 
aid of the formula of ScHUBERT : - 
Eer dd 
which relates to a set of o' pairs of points. We can now indeed 
obtain such a set by conjugating on each line of the regulus each 
of the six points £**, regarded as a point p, to the five others, which 
are then named q ; each line of this kind bears then thirty pairs, because 
each of the six points of k'* tying on it can be conjugated succes- 
') ScavBeRT |. c. p. 44. 
