( 580 j 
only of one generator situated in the plane of this section, i. e. of 
the line joining the w-fold point and the v-fold point of the section; 
the section C*++! being itself a simple curve of the surface, the 
order of this skew surface is wtv4-2; and m and » are (u t-1)- 
fold and (y-H1)-fold lines of it. By another plane section Du+++1 of 
Sety+? it is met in 2uy+-ye+y-+1 points belonging neither to m, nor 
to n, nor to C##+1 , So the multiple lines m and » cut 2uvt+u+y+1 
simple lines of S#t+’+1 , 
Evidently the last number is modified if a third multiple line 
cutting m and x presents itself. So a A-fold line J reduces this num- 
ber of single lines by A? as it is also a A-fold line of the indicated 
skew surface. 
S 7. If e. g. the surface S° has two double lines m and 2, we 
find 13 single lines cutting these. Each of the lines m, n counting 
for 16 points in the determination of the surface, still 23 points can 
be chosen at random; so we can make S® to pass through three 
lines ej, ca, ez crossing m, x?) Then the relation of position between 
the different lines can be indicated by the following table: 
C‚. meets dì d, ds dy dy 
Ca eerd Sd dz de ds 
es 2 de dy de de dy 
al meet dh dy “dy he Ue dy dy Ms Wy A ae 
> 
If S® still admits of a threefold line /, this line replaces 9 of the 
13 lines d. But then each of the lines m, » counts for a couple of 
lines cutting the threefold line d. Then the curve of double curvature 
considered in § 1, by the occurrence of the double points M,, WN, 
of the plane curve C5,, is of order four; as it meets m, n the 
surface S° can only have three more pairs of lines meeting /. Their 
position with respect to the lines d is indicated by the table: 
d, rests ON @y9, M3, Wy4; 
ds ” 95 boss Das 5 
ds » Day Ai3, Oos ; 
dy ” Daas boa, 4+ 
1) Also four lines cx forming a skew quadrilateral can be assumed. 
