1452 
hk with the point of contact of ¢; if ¢ now coincides with /,, the 
point of contact remains definite, the intersection with /, does not, 
and so we can connect the point of contact with any point of 4%, 
in order to find always a straight line, which does belong to the 
“true” surface ‘/; this is the reason why /, not belongs to {2 
either. To 2 does belong, however, the line connecting the point 
of contact of Ue and k’ with Z,, as is easily to be seen if the 
tangent ¢ is made to approach to /,. At the same time we are 
then convinced that at the limit ¢#vo generatrices coincide in this 
line, according to its two intersections with £, so that it is a 
double generatrix; but we should moreover consider that even as a 
double generatrix it is to be taken twice, as / has in common with 
Ll, two infinitely near points, and for one point the same obtains 
that obtains for the other; we may say that it is a double torsal 
line, whereas the tangent-plane coincides both times with «,. 
This becomes still more evident if we just consider an ordinary 
intersection S, of £? with 7. By causing a point P of k” to 
approach to S, we are at once convinced that SZ, is a double 
generatrix of £2, and again a double torsal line, with a tangent- 
plane, however, that contains the tangent S, at 4’; if now two 
points S get to lie infinitely near, two double generatrices get to lie 
infinitely near. 
These considerations enable us moreover to control the order 7 
of 2 arrived at above by means of the plane at infinity of space. 
The intersection of this plane with @ consists viz. of the following parts: 
a. the 26 double generatrices lying in pairs infinitely near, arising 
from the o points of contact of 4% with /, ; 
b. the u— 2e — 26 double generatrices arising from the simple 
intersections of k” with 1; 
c. the conic &,. This is a (p —o)-fold curve of the surface, for 
if an arbitrary point A, of £ is connected with Z,, and the con- 
necting line is made to intersect /,, there pass through the inter- 
section »—o tangents at 4”, whose point of contact does not lie 
at infinity, consequently pass through A, »—o generatrices of 22. 
By means of the plane at infinity of space we find therefore for 
the order of 2: 
mda 2(u— 2e—26)+ 2(p — 0) = 2 (u + vp — BE — 0). 
As the points J,,, 4, 
Loo? 20 
lie on k’, even as e-fold points, it might 
be expected that the two straight lines Z, /,,, Zl, lay also on 2; 
this, however, is not so, and that because these lines touch 42 instead 
of cutting it. If along one of the ¢ branches of &” passing through 
