(581 >) 
§ 8. By supposing the threefold line (2) and the double line (m) 
of S5 to coincide with the axes of Z and Y, it is immediately seen 
that these lines count for 28 and 19 points in the determination of 
the surface and that still 11 points more can be chosen arbitrarily. 
Instead of a second double line » the four lines ej, co, cz, ¢4 resting 
on m can be assumed at random. Then we find that / is cut by 
seven pairs of lines, the position of which with respect to the lines 
ce, is as follows: 
C, cuts Mes, Miza, Arsa) 41, Ae, Gis, Mas 
Co» 93 Aaa, Dg, Maga Aro, O24, Das; 
C3 498, Bg, 134, ana, ba, A3, Dag; 
Can as Qigay A134) Coza, O34, Dou) Cia: 
S 9. In order to obtain a group of surfaces with two multiple 
lines we first consider the locus of a plane curve C* cutting n-times 
the right line 7 and once each line of a given set of  n (n+3) lines 
ez crossing J. The order of this surface is equal to the number of 
3 1 
curves C” lying in planes through / and resting on a er) (n-+-2) 
lines c. To find this number f(x) we suppose that (+1) of these 
lines c meet /; then the plane passing through / and any of these 
(n-+1) lines contains a single curve C” satisfying the conditions 
and is evidently to be counted n-times. All the remaining curves C” we 
are in search of break up into / and a curve ("—! resting on 
=" (x +1) lines e. So the relation /(n) =» (n-+1) + f(n—1) holds; 
moreover /(1) representing the number two of the common transver- 
sals of four crossing lines, we find immediately / (x) = 7 n (n+1)(n+2) 
for the order of the loeus of the curve C” meeting » times the line 
l and onee each of the» (n+3) given lines cr. 
Considering curves Cr in planes through / which pass «-times through 
the point of intersection of the plane with m and once through each of 
: anid 1 
the points of intersection of the plane with ue (n + 3) — a ALE) 
lines c, we find for the order p (x, 4) of the locus the relations 
p(n, 4) =n(n +1) + p (rn — 1, ), ete. until we get p (u + 2, 4) 
= (u + 2) (u + 3) + Pp (u + 1, 4). 
Here plu 1l,g) is the number of plane curves Cet! with a 
u-fold point on m, cutting (2 + 3) given lines c. By supposing 
that zw — 2 of these lines rest on / the curves C*+! break up into 
l and a C# with «-fold point on m cutting (w + 1) lines ¢, i, e, 
