724 
now with / make its appearence? An arbitrary point is gene- 
rated when in the plane / through that point and / a generatrix 
of 2°° and a conic of @* touch each other; so a point on lis gene- 
rated when in a certain plane 4 through / a generatrix of @* and 
a conic of @* touch each other exactly on /; then through the point 
of contact, however, pass two coinciding tangents of the conic, thus 
two coinciding complex rays; or, in a better wording, whilst in an 
arbitrary plane 2 through each of the 14 points of / lying at the 
same time on generatrices of @?° two complex rays pass one of which 
does not belong to £2, in the case under discussion the last 
ray coincides with the former, so tbat it might look as if here a 
torsal line of the third kind was generated; but it would have to be 
possible to show that in the plane through / and such a line only 
twelve other generatrices of 2*° were situated, or that whilst tend- 
ing to such a plane two generatrices were tending to each other, 
for whieh there is no reason whatever; so we conclude that 2?° 
does not possess torsal lines of the third kind, and we shall find this 
conclusion justified in future in different moments. 
16. In a plane 2 through / lie fourteen generatrices of 2%; 
through each of the points £ in which these generatrices intersect 
l five other generatrices pass which in general determine with / 70 
different planes; we shall conjugate these to 2. In this manner the 
planes through / are arranged in a symmetrical correspondence of 
order 70; we wish to submit the 140 double planes d of this 
correspondence to a closer investigation. Such a plane is evidently 
generated if for a certain point L of / two of the 6 generatrices s 
through that point lie with / in the same plane; the point Z is then 
evidently at the same time a point of the nodal, curve of 2*° lying 
on /, for this double curve is the locus of the points of intersection 
of all generatrices lying in a plane 4 through /. We shall now, 
however, show that each suchlike plane as a matter of fact represents 
two coinciding double planes. Let us assume to that end a plane 4 
in which two generatrices s,,s, are lying, cutting / in two points 
L,, LZ, lying close together. Through each of these last pass five 
generatrices not lying in 2, and that in such a way that one of the 
generatrices through Z, lies in the vicinity of s, and inversely, whilst 
the remaining ones lie two by two in each other's vicinity. If 
we allow 2 to transform itself gradually into d, then that one gene- 
ratrix through Z, coincides with s,, and inversely, whilst the remaining 
ones coincide two by two in four double planes of the second kind 
