( 582 ) 
into 7 and a star of u rays passing through this m-fold point. So 
one of the curves Cé in question is found by uniting a common 
transversal of /, m and any pair of the (u + 1) lines c by a plane 
and joining the point of intersection of this plane with m to the 
points where this plane meets the remaining (g — 1) lines c. So 
g(etilw=(e+t)(e+2)4+ (4+ 1u =d. And this 
; 1 
number being inferior to f(t + 1) by (u +1) wu — 1), we find 
1 1 
Blast Pod ne 5 (u F1) ge (u — 1). Therefore 
the locus under investigation is a surface of order p(n, u) with a 
; : : l 1 
ge-fold line m, al (n, se) — n]-fold line land aa (n + 3) — at (u + 1) 
single lines. 
§ 10. For »=8, w= 2, the obtained surface is an S!® with a 
15-fold line /, a double line m and six single lines cj. 
By cutting it by a curve of double curvature 2? of which J,m, 
€1,€g,¢3 are double secants it is seen that a surface S! is obtained 
if the director line ce, is replaced by the curve of double curvature 23. 
Furthermore a surface S'° and a surface SC are found if ¢; is 
also replaced by a curve of space #,°, or if c; and cy are replaced 
by curves of space Lj’, Ro? having J, m, c,¢9,¢3 for douole secants. 
In the supposition that these two curves /°, 2,° have a point in 
common the obtained surface is S® with two double lines J, m and 
three single lines ¢),¢9,¢; (compare § 7). It is clear that surfaces 
with two multiple lines /, m and any number of single lines c alter- 
1 1 
nating from unity to 5 (n° + 3n — 2) — > U (4 + 1) can be obtained 
in the same way. 
$ 11. By choosing the w-fold line m of a surface S* for axis 
of z it is easily seen that the assumption of a w-fold line counts for 
1 : cee 
AE J1)(3n — 2Ww +5) points. So a surface St admitting of a 
u-fold line m and a v-fold line # can still be made to pass through : 
5 (ted?) (ety $8) (why $4) — 1 (ge HI) (e487 48) — 
1 
Er Buty +8) = er dS Hv + IJ 
points, For v = 4 this is 24? +- Ope + 3 = (u +1) (2u + 2) + (24+ 1), 
