(213 ) 
bearers of tangent planes and double points, whilst each double six 
of the surface of order three transforms itself into a double six of 
the surface of class three. In the second place — and with more 
right — we see therefore in the notation with two indices applied 
above to the lines of the configuration the notation of the fifteen lines 
of three-dimensional space forming with a double six the 27 right 
lines of a surface of class three. 
If we project the fifteen right lines of the configuration out of 
any point P, neither with two crossing lines of the fifteen in the 
same space nor with two cutting lines of the fifteen in the same 
plane, on any space S; not containing this point P, we then have 
in Ss fifteen lines ap, which are really bearers of tangent planes 
of a single surface of class three. By polarization of a wellknown 
proof we find namely, that the lines 
' 
| a 12 a 34 a 56 
| 1 ’ 
a 45 @61 a 93 
, 1 1 
a 36 a 5 a 14 
as the lines connecting two triplets of points representing degener- 
ated surfaces of class three form the developable surface degenerated 
into nine pencils of planes enveloping a tangential pencil of surfaces 
of class three. And now the surface belonging to that tangential 
pencil touching at the same time the plane passing through a'j, 
and a's, has with each of the six pencils of planes round the 
remaining lines 
four planes in common, so that this surface has all the fifteen 
lines a’; as bearers of tangent planes. 
But this same surface of class three is moreover connected in a 
simple manner with the sextuples of the conjugate lines which in 
future we shall briefly indicate by the sign (x), where 7 stands for 
the common index of the lines. For, this surface contains besides 
the fifteen lines a' still twelve lines forming a double six. And if 
we indicate this double six in connection with these lines a’ in the 
manner customary with the surface of order or class three by 
14* 
