1455 
L 
1x 
a point P with tangent ¢ is made to approach to 4/1» direct 
contemplation teaches that as the intersections with #7 of the 
line connecting Z, with the point at infinity of ¢, simultaneously 
with P approach to /,,, the two generatrices passing through 
P approach to limit positions not coinciding with /,, 7. Through 
1, 45, pass therefore every time r—2e—o generatrices lying in 9, 
and 2e others not lying in ? but neither passing through 7, ; together 
therefore »—o, as well as through any other point of 4. 
Of the w—2e nodal edges passing through Z, ~—2e—2e lie iso- 
lated, while the 26 remaining ones coincide in pairs; this influences 
the multiplicity of the point 7, considered as a point of the surface. 
As namely in general through a point where two nodal edges, or 
more generally two nodal lines meet, four sheets of the surface pass, 
and this point consequently becomes a quadruple point for the surface, 
there pass through the intersection of two infinitely near nodal 
edges only two sheets, viz. simply those two that touch along those 
edges ; the conseqence of this is that an arbitrary straight line passing 
through Z, does not cut the surface there in 2 (u—?2e), but only 
in 2 (wu — 2e — 20) + 20 = 2 (u — 2e — 0) points, so that Z,, is for 
our surface a 2(u—2e—o)-fold point. The 5 pairs of coinciding 
nodal edges are torsal lines of 2, and they are to be counted twice, 
because two sheets of $2 touch each other along each of them. 
As the order of {2 is equal to 2u + 2v — 4e — 25, and a straight 
line passing through Z, has, in this point only, already 2u—4e—2v7 
points in common with £2, only a number 2r remains for the inter- 
sections not lying in this point; they lie in pairs symmetrically in 
regard to the plane of 4, and are represented by the r-circles, which 
may evidently be described round the foot point of the straight line 
as centre in such a way that they intersect 4” perpendicularly. 
At first sight it is somewhat striking that the order of the non- 
developable surface which is under present observation, corresponds 
exactly with that of the developable surface in the treatise quoted 
in the Introduction, which surface we defined at the time as the 
common circumscribed developable surface of £” and 4°, ; the peculiarity 
of this phenomenon disappears, however, if we observe that we 
might have constructed this developable surface as well by applying 
the construction which we now apply to the tangents of hk’, to the 
normals of £?, which we have not done, however, as in that way 
its character as a developable surface becomes less prominent. 
§ 3. In § 1 we found that 4” is a nodal curve for £2, and we 
will now investigate how the two sheets of the surface passing 
